메트리저블 위상 벡터 공간

수학의 기능 분석과 관련 영역에서, 측정 가능한 (resp. phasmetrizable) 위상 벡터 공간(TVs)은 위상이 미터법(resp. phasometric)에 의해 유도되는 TVS이다.LM-공간은 지역적으로 볼록한 전이 가능한 TVS 시퀀스의 유도 한계다.

유사 측정 및 측정 기준

집합 $X{\displaystyle }$ ${\displaystyle X}$의 유사 측정값은 ${\displaystyle \mathrm{다음}}$ 속성을 충족하는$$ 지도 d${\displaystyle }$: ${\displaystyle X}$ ${\displaystyle \times }$ ${\displaystyle X\mathbb{}}$→ R ${\$ X$\오른쪽 화살표 \mathb$ {$R}}$이다.

1. ${\displaystyle d}$( ${\displaystyle x}$, x${\displaystyle }$)${\displaystyle }$= ${\displaystyle \text{모든}}$ ${\displaystyle x}$ ∈에 ${\displaystyle \text{대해}}$ 0 = 모든 x ${\displaystyle \in }$ X에$$대해 {\$displaystyle d(x,x)=$0{\$text{ 모든 }x\in X$
2. 대칭: $d{\displaystyle }$( ${\displaystyle x}$, ${\displaystyle y}$)${\displaystyle }$= ${\displaystyle d}$ (${\displaystyle y}$ , ${\displaystyle x}$) = ${\displaystyle \text{모든}}$ ${\displaystyle x}$, ${\displaystyle y}$ ${\displaystyle \in }$ X${\displaystyle d(x,y)=d(y,x){\text{{{}x,y\in$ X$}$;
3. Subaditivity: ${\displaystyle d}$( ${\displaystyle x}$, ${\displaystyle z}$) ${\displaystyle \le }$ d${\displaystyle }$( x${\displaystyle }$, ${\displaystyle y}$)${\displaystyle }$+ ${\displaystyle d}$( y${\displaystyle }$, ${\displaystyle z}$) ${\displaystyle \mathrm{모든}}$ ${\displaystyle \text{x}}$, ${\displaystyle y}$ , ${\displaystyle z\in }$ X${\displaystyle }$. ${\displaystyle d(x,z)\leq d(x,y)+d(y,z){\text{}x,y,z\in X.}$

가성비는 다음을 만족하는 경우 미터법이라고 한다.

4. 불분명한 개체의 ID: ${\displaystyle \mathrm{모든}}$ ${\displaystyle x}$, y ${\displaystyle \in }$ X${\displaystyle }$, ${\displaystyle x,y\in X,}$에 대해, $d{\displaystyle }$( x${\displaystyle }$,${\displaystyle y}$) = 0$${\$displaystyle d(x,y)=0}$이면${\displaystyle }x{\displaystyle }$ = ${\displaystyle y.}${\$displaysty$x=$y.$$}$

초고속도계

$X{\displaystyle }$ ${\displaystyle X}$의 가성 $d{\displaystyle }${\$displaystyle d}$은(는) 다음을 만족하는 경우 초밀도계 또는 강한 가성계라고 한다$$.

5. $강{\displaystyle }$/${\displaystyle \mathrm{초계}}$ 삼각형 불평등: d${\displaystyle }$( x${\displaystyle }$, y${\displaystyle }$) ${\displaystyle \le }$ ${\displaystyle \max }${ d (${\displaystyle x}$,${\displaystyle z}$) ,${\displaystyle d}$ ( y${\displaystyle }$, ${\displaystyle z}$) ${\displaystyle \}}$ ${\displaystyle 모든}$ x${\displaystyle }$, $x$ .$$\\d$(x, z)\leq \max\{d(x,z),d(y,z)\}{\text{}{}}}$ }.$x,y\in.$

가성공간

A pseudometric space is a pair ${\displaystyle (X,d)}$ consisting of a set ${\displaystyle X}$ and a pseudometric ${\displaystyle d}$ on ${\displaystyle X}$ such that ${\displaystyle X}$'s topology is identical to the topology on ${\displaystyle X}$ induced by ${\displaystyle d.}$ We유사 측정 공간$({\displaystyle X}$, d${\displaystyle }$) ${\displaystyle(X,d)}$을(를) 메트릭 공간(resp)이라고 부른다.$d{\displaystyle }$ ${\displaystyle d}$이(가) 메트릭인$$ 경우 초경량 공간.극초단파 측정학

유사 측정에 의해 유도된 위상

$d{\displaystyle }$ ${\displaystyle d}$이(가) $설정$된 X ${\displaystyle X}$의 유사 측정값인$$ 경우, 열린 볼 모음:

으로서 z{z\displaystyle}X{X\displaystyle}과 r를 혼동하고,는 긍정적인 실수를 0개 이상의{\displaystyle r>0}이 솟아나고, X{X\displaystyle}에 X{X\displaystyle}에 d{\displaystyle d}-topology 또는 합치 토폴로지 d에 의해 유도된 토폴로지에 대한 기초를 형성한다. {\di$splaystyle d.$$}$

규약:${\displaystyle }$( ${\displaystyle X}$, d${\displaystyle }$) ${\d$ ${\displaystyle \mathrm{디스플레이}}$ 스타일 $(X,d)}$이$($가) 유사 측정 $공간$이고 X ${\displaystyle$ X}이(가$)$ 위상학적 공간으로 취급되는$$ 경우, 달리 $명시$되지$않는$ 한, X {\displaystyle $X}$이 $d{\displaystyle }$. ${\displaystytyle$ d에 의해 유도된 위상과 함께 귀속된다고 가정해야 한다.$}$

가성측정 가능 공간

A topological space ${\displaystyle (X,\tau )}$ is called pseudometrizable (resp. metrizable, ultrapseudometrizable) if there exists a pseudometric (resp. metric, ultrapseudometric) ${\displaystyle d}$ on ${\displaystyle X}$ such that ${\displaystyle \tau }$ is equal to the topology induced by ${\dis$$연극을$$하다$$}$^[1]

위상학 그룹의 유사측정학 및 값

가법 위상학 그룹은 그룹 위상이라고 불리는 위상을 부여받은 첨가물 그룹이며, 그 아래에 덧셈과 부정이 연속적인 연산자가 된다.

실제 또는 복잡한 벡터 공간 $X{\displaystyle }$ ${\displaystyle X}$의 위상 $topology{\displaystyle }$ ${\displaystyle \tau }$을(를) 벡터 추가 및 스칼라 곱셈의 연산을 연속적으로 하는 경우($즉{\displaystyle }$,$$X {\$displaystyle X$}을(를 위상 벡터 공간으로 만드는$$ 경우) 위상 또는 TVS 위상이라고 한다$$.

모든 위상 벡터 공간(TV) $X{\displaystyle }$ ${\displaystyle X}$은(는) 가법역학 위상 그룹이지만 $X{\displaystyle }$ ${\displaystyle X}$의 모든 그룹 위상이 벡터 위상인 것은$$ 아니다.추가 및 부정을 연속적으로 하지만 벡터 공간 $X{\displaystyle }$ ${\displaystyle X}$의 그룹 토폴로지가 스칼라 곱셈을 연속적으로 만들지 못할$$ 수 있기 때문이다.예를 들어, 비삼각 벡터 공간의 이산 위상은 덧셈과 부정을 연속적으로 만들지만 스칼라 곱셈을 연속적으로 만들지는 않는다.

번역불변성 유사성

$X{\displaystyle }$ ${\displaystyle X}$이(가) 가법 $그룹$인 경우 X ${\displaystyle X}$의 d ${\displaystyle d}$은(는) 변환 불변성이거나 다음 등가 조건 중 하나를 만족하는 경우 불변성이라고 말할 수 있다.

1. 변환 호출: ${\displaystyle d(}$ ${\displaystyle x}$+ z${\displaystyle ,}$ ${\displaystyle y}$+ ${\displaystyle z}$)${\displaystyle }$= ${\displaystyle \text{모든 x}}$, ${\displaystyle y}$, z ${\displaystyle \in }$ ${\displaystyle X}${\$displaystyle d(x+z,y+z)=d(x,y){\text{{}x,y,z\in$ X$\};$
2. ${\displaystyle d(x,y)=d(x-y,0){\text{모든 }x,y\in.}$

Value/G-seminorm

$X{\displaystyle }$ ${\displaystyle X}$이(가) 위상학 그룹인 $경우{\displaystyle }$ X ${\displaystyle$ X$}($G는 Group을 나타냄)의 값 또는 G-세미놈은 다음과 ^[2]같은 속성을$$ 가진 실제 값${\displaystyle }p{\displaystyle }$ :${\displaystyle X}$ → ${\displaystyle R\mathbb{}}$ → R ${\$이다$.$

1. 비음성: $p{\displaystyle }$≥ ${\displaystyle 0.}$ ${\displaystyle p\geq.}$
2. Subadditive: ${\displaystyle p}$( ${\displaystyle x}$+ y${\displaystyle }$)${\displaystyle \le }$ ${\displaystyle p}$( ${\displaystyle x}$)${\displaystyle }$+ p ( ${\displaystyle y}$ ) + p ( y ) 모든 x , ${\displaystyle y)X}$ ${\displaystyle p(x+y)\leq p(x)+p(y){\text{{}모든 }x,y\in$ X$\};;$
3. ${\displaystyle p(0)=0..}$
4. 대칭: $p{\displaystyle }$ ( -${\displaystyle x}$)${\displaystyle }$= ${\displaystyle p}$ (${\displaystyle }$x ) = ${\displaystyle \text{모든}}$ ${\displaystyle x(}$ ${\displaystyle X}$. ${\displaystyle p(-x)=p(x$){\$text{모든 }x\in X.}$

추가 조건을 만족하는 경우 G-seminorm을 G-norm이라고 부른다.

5. 총계/양수 한정자:$p{\displaystyle }$( ${\displaystyle x}$)${\displaystyle }$= ${\displaystyle 0}$ ${\displaystyle p(x)=0}$이면 $x{\displaystyle }$= ${\displaystyle 0.}$ ${\displaystyle$ x$=0.$$}$

값의 속성

$p{\displaystyle }$ ${\displaystyle p}$이(가) 벡터 공간 $X{\displaystyle }$ ${\displaystyle X}$의 값인$$ 경우$$:

• ${\displaystyle p(x)-p(y) \leq p(x-y){\text{모든 }x,y\in.}$^[3]
• ${\displaystyle p(nx)\leq np(x)}$ and ${\displaystyle {\frac {1}{n}}p(x)\leq p(x/n)}$ for all ${\displaystyle x\in X}$ and positive integers ${\displaystyle n.}$^[4]
• 세트${\displaystyle }${ ${\displaystyle x}$ ${\displaystyle \in }$ ${\displaystyle X}$: ${\displaystyle p}$ (${\displaystyle }$x${\displaystyle }$)${\displaystyle }$= ${\displaystyle 0}$ ${\displaystyle \}}$ ${\displaystyle \{x\in X:p(x)=0\}}$은 $X{\displaystyle }$. ${\displaystyle$ X$.}$의 가법 하위 그룹이다.

위상군 등가성

유사측정 가능한 위상학군

벡터 위상을 유도하지 않는 불변 가성계

Let ${\displaystyle X}$ be a non-trivial (i.e. ${\displaystyle X\neq \{0\}}$) real or complex vector space and let ${\displaystyle d}$ be the translation-invariant trivial metric on ${\displaystyle X}$ defined by ${\displaystyle d(x,x)=0}$ and ${\displaystyle d(x,y)=1\text{for all}x,y\in X}$${\displaystyle d(x,y)=1{\text${{{\text{$모든 }x,y\in X}$에 ${\displaystyle \mathrm{대해}}$, x ${\displaystyle \ne }$ $y{\displaystyle .}$ ${\displaystyle$ x$\neq y.}$$$ $d{\displaystyle }$ ${\$$displaystyle$ $d$$}$이(가) X ${\displaystyle$ X$}$에 유도하는$$$$ 위상 $\tau {\displaystyle }$ ${\$$displaystyle$$\tau$$$}은(${\displaystyle }$X ,${\displaystyle }$${\displaystyle \tau }$ ${\displaystyle \tau }$\ \ ${${ { \ \ \ \ \)은 이산 위상이며, 이 위상은 (${\displaystyle X}$ ,${\displaystyle }$$x{\displaystyle }$ $\$ } } \ } } }$$} } } } } } } } } } } } } } } } $}$ } } }$splaystyle(X,\tau )}$은(는) 연결이 끊어졌지만 모든 벡터 위상이 연결되어 있다.실패한 것은 메스커 곱셈이${\displaystyle }$(${\displaystyle X}$, ${\displaystyle \tau }$ )${\displaystyle }$.${\displaystyle (X,\tau$) 에서 연속되지 않는다는 것이다.$}$

이 예는 번역-인바리어스(의사) 측정학으로는 벡터 위상(vector topology)을 보장하기에 충분하지 않다는 것을 보여주는데, 이것은 우리가 편집증과 F-세미나(f-seminorms)를 정의하도록 이끈다.

가법순서

A collection ${\displaystyle {\mathcal {N}}}$ of subsets of a vector space is called additive^[5] if for every ${\displaystyle N\in {\mathcal {N}},}$ there exists some ${\displaystyle U\in {\mathcal {N}}}$ such that ${\displaystyle U+U\subseteq N.}$

위의 모든 조건은 결과적으로 위상이 벡터 위상을 형성하는데 필요한 것이다.집합의 가법 시퀀스는 음이 아닌 연속 실제 값 하위 가법 함수를 정의하는 특히 좋은 특성을 가지고 있다.이 기능들은 위상학적 벡터 공간의 많은 기본적 특성을 입증하는 데 사용될 수 있고 또한 이웃의 셀 수 있는 기반을 가진 하우스도르프 TVS가 메트리징 가능하다는 것을 보여준다.다음과 같은 정리는 교번적 첨가물 위상학 그룹에 더 일반적으로 적용된다.

파라노름스

If ${\displaystyle X}$ is a vector space over the real or complex numbers then a paranorm on ${\displaystyle X}$ is a G-seminorm (defined above) ${\displaystyle p:X\rightarrow \mathbb {R} }$ on ${\displaystyle X}$ that satisfies any of the following additional conditions, each of which begins with "for allsequences ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }}$ in ${\displaystyle X}$ and all convergent sequences of scalars ${\displaystyle s_{\bullet }=\left(s_{i}\right)_{i=1}^{\infty }}$":^[6]

1. Continuity of multiplication: if ${\displaystyle s}$ is a scalar and ${\displaystyle x\in X}$ are such that ${\displaystyle p\left(x_{i}-x\right)\to 0}$ and ${\displaystyle s_{\bullet }\to s,}$ then ${\displaystyle p\left(s_{i}x_{i}-sx\$$오른쪽)\0.}$
2. 두 가지 조건 모두:
   • if ${\displaystyle s_{\bullet }\to 0}$ and if ${\displaystyle x\in X}$ is such that ${\displaystyle p\left(x_{i}-x\right)\to 0}$ then ${\displaystyle p\left(s_{i}x_{i}\right)\to 0}$;
   • $p{\displaystyle (}$ ${\displaystyle {}_{}}$ ${\displaystyle {\bullet }_{}}$)${\displaystyle }$→ ${\displaystyle 0}$ ${\displaystyle p\left(x_{\put }\오른쪽$)\$p$$($$s$ ${\displaystyle {}_{}}$i${\displaystyle {}_{}}$)${\displaystyle }$→ $모든$ 스칼라 s. ${\displaystyle$ $s$$.}$에 대해$$ 0${\displaystyle (s}$ x ${\displaystyle i_{}}$ ) →$0($ x i}\displaystystylef${\displaystyle .}$
3. 두 가지 조건 모두:
   • $일부{\displaystyle }$ $스칼라{\displaystyle }$ s${\displaystyle {}_{}\{\backslash }$$displaystyle s}$에 대해 p${\displaystyle ({}_{}}$ $∙$ ) → $(x_$${\put$ $}\$$right$)$및$ s$$ ${\displaystyle {}_{}}$${\displaystyle s_{})}$ → s$(s$ i x$$i )${\displaystyle }$→ 0$(\displaystyle p\p\s)$→ $0(s_{$i}x_{$i$}\$rig$}\right$)\backslash \}$
   • $s{\displaystyle {}_{}}$ → ${\displaystyle 0}${\$displaystyle s_{\bullet }\\}$을(를) 0으로(를) 지정하면$$ $p{\displaystyle ({}_{}}$ i x${\displaystyle }$)${\displaystyle }$→ ${\displaystyle \text{모든}}$ $x$ ${\displaystyle X}$에 대해$$으로($s_{i}x\right)\to$ 0${{\text}$
4. 별도의 연속성:^[7]
   • 일부 스칼라 $s{\displaystyle }$ ${\displaystyle$ $s$$}$에 대해 $s{\displaystyle {}_{}}$ ∙${\displaystyle {}_{}}$→ s ${\$$displaystyle$ $s_{\bullet }\to s}$이면 p (${\displaystyle s{x}_{}}$ ${\displaystyle {}_{}}$- ${\displaystyle s}$)${\displaystyle }$→ ${\displaystyle \mathrm{매}}$ $x{\displaystyle }$ x${\displaystyle x}$X에 대해$$ p$\left(sx_{i}-sx$$\right$$)\to 0$$.$
   • $s{\displaystyle }$ ${\displaystyle s}$이(가) 스칼라, ${\displaystyle x}$ ${\displaystyle \in }$ X${\displaystyle }$, ${\$$displaystyle x\$$in$ X${\displaystyle ,\}}$및 $p$ ( x${\displaystyle {}_{}}$- $x_{i}-x\right)$ → $p$${\displaystyle (}s$ x i -${\displaystyle {}_{}s}$ )${\displaystyle }$→ 0(${\displaystyle s}$ ${\displaystyle {}_{}}$ ${\displaystyle i_{}}$ - s${\displaystyle )}$ → $($s x - s) ${\displaystylef p\p$ ($s_{i-s-s-s-s-s-s-s-sx_{sx_${sx$\x$\rimef) $to$ 0$}$

편집광은 추가로 다음을 충족한다면 총체라고 불린다.

• $총계{\displaystyle }$/${\displaystyle \mathrm{양수}}$ 한정자: p${\displaystyle }$( x${\displaystyle }$)${\displaystyle }$= ${\displaystyle 0}$ ${\displaystyle p(x)=0}$은 $x{\displaystyle }$= ${\displaystyle 0.}$ ${\displaystyle$ x$=0$을 의미한다$.$$}$

파라노몬의 속성

If ${\displaystyle p}$ is a paranorm on a vector space ${\displaystyle X}$ then the map ${\displaystyle d:X\times X\rightarrow \mathbb {R} }$ defined by ${\displaystyle d(x,y):=p(x-y)}$ is a translation-invariant pseudometric on ${\displaystyle X}$ that defines a vector $X{\displaystyle }$. ${\displaystyle$ X$.}$의 토폴로지

$p{\displaystyle }$ ${\displaystyle p}$이(가) 벡터 공간 $X{\displaystyle }$ ${\displaystyle X}$의 편집증인 경우$$$$:

• 세트${\displaystyle }${ ${\displaystyle x}$ ${\displaystyle \in }$ ${\displaystyle X}$: ${\displaystyle p}$ (${\displaystyle x}$)${\displaystyle }$= ${\displaystyle 0}$ ${\displaystyle \}}$ ${\displaystyle \{x\in X:p(x)=0\}}$은 $X{\displaystyle }$. ${\displaystyle$ X$.}$의 벡터 하위 공간이다$$.
• ${\displaystyle }$$p{\displaystyle }$${\displaystyle }$( ${\displaystyle x}$+ ${\displaystyle n}$)${\displaystyle }$= ${\displaystyle p}$( x${\displaystyle }$) = 모든 ${\displaystyle x}$, ${\displaystyle n}$ ${\displaystyle \in }$ X ${\displaystyle p(x+n)=p($${\displaystyle x}$$){\text{{},$ p $($ ) = $0$${\displaystyle \mathrm{.}}$ ${\displaystyp p(n)=0$. {\displaystyp p$($n)=0.$}$^[8]
• If a paranorm ${\displaystyle p}$ satisfies ${\displaystyle p(sx)\leq s p(x){\text{ for all }}x\in X}$ and scalars ${\displaystyle s,}$ then ${\displaystyle p}$ is absolutely homogeneity (i.e. equality holds)^[8] and thus ${\displaystyle p}$ is a seminorm.

편집증적 괴물의 예

• If ${\displaystyle d}$ is a translation-invariant pseudometric on a vector space ${\displaystyle X}$ that induces a vector topology ${\displaystyle \tau }$ on ${\displaystyle X}$ (i.e. ${\displaystyle (X,\tau )}$ is a TVS) then the map ${\displaystyle p(x):=d(x-y,0)}$$({\displaystyle X}$, ${\displaystyle \tau }$) ${\displaystyle (X,\tau )}$에 대한 연속 편집증적 편집증적 편집증적 정의를 정의한다$.$ 더욱이 이 편집증적 $p{\displaystyle }$ ${\displaystyle$ $p$$}$이(가) $X{\displaystyle }$ ${\displaystyty$ X$}$에서 정의하는$$ 위상은$$ $\tau {\displaystyle .}$ ${\displaystyle \tau.}$이다.
• $p{\displaystyle }$ ${\displaystyle p}$이(가) $X{\displaystyle }$ ${\displaystyle X}$의 편집자라면 지도$$ $q{\displaystyle }$(${\displaystyle x}$ ) ${\displaystyle :=}$ p${\displaystyle }$( ${\displaystyle x}$)${\displaystyle }$/${\displaystyle }$[ ${\displaystyle 1}$+ ${\displaystyle p}$( x${\displaystyle }$)${\displaystyle }$.${\displaysty q:=p(x)/[1+p(x)]$도 마찬가지다.$}$^[8]
• 모든 양성 스칼라 다발성 편집증(resp. total panalorm)은 다시 그런 편집증이다.
• 모든 세미몬은 편집광이다.^[8]
• 벡터 서브 스페이스에 대한 편집광(resp. total panalorm)의 제한은 편집광이다.^[9]
• 두 명의 편집광의 합은 편집광이다.^[8]
• $p{\displaystyle }$ ${\displaystyle p}$ 및$$ $q{\displaystyle }$ ${\displaystyle q}$이(가) X ${\displaystyle \{\backslash }$$displaystyle X}$에서 편집광인 경우$$${\displaystyle }$(p$$${\displaystyle q}$)${\displaystyle :=}$ ${\displaystyle \inf \underset{}{}}${${\displaystyle \underset{}{p}}$ ${\displaystyle (y}$ ${\displaystyle )}$+${\displaystyle q}$ (${\displaystyle z}$)${\displaystyle }$: ${\displaystyle x}$= ${\displaystyle \text{y}}$ = ${\displaystyle y}$ = ${\displaystyle y,}$ ${\displaystyle z}$ ${\displaystyle \in }$ ${\displaystyle X}$ ${\displaystyle \}}$.$${\$displaystylease (p\wedge q)(x):$$=\inf _{}\{p(y)+q(z):$$x=y+z{\text{ with }}y,z\in X\}.}$ Moreover, ${\displaystyle (p\wedge q)\leq p}$ and ${\displaystyle (p\wedge q)\leq q.}$ This makes the set of paranorms on ${\displaystyle X}$ into a conditionally complete lattice.^[8]
• 다음의 각 실제 가치 지도는 $X{\displaystyle }$ ${\displaystyle :=}$ ${\displaystyle {\mathbb{}}^{}}$ ${\displaystyle {\mathrm{}}^{}}$ ${\$ X$:=\mathb {R} ^{2$
  • $(x,y)\mapsto x }$
  • $#\displaystyle (x,y)\mapsto x + y }$
• The real-valued maps ${\displaystyle (x,y)\mapsto {\sqrt {\left x^{2}-y^{2}\right }}}$ and ${\displaystyle (x,y)\mapsto \left x^{2}-y^{2}\right ^{3/2}}$ are not a paranorms on ${\displaystyle X:=\mathbb {R} ^{2}.$$}$^[8]
• If ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ is a Hamel basis on a vector space ${\displaystyle X}$ then the real-valued map that sends ${\displaystyle x=\sum _{i\in I}s_{i}x_{i}\in X}$ (where all but finitely many of the scalars ${\displaystyle s_i}$${\displaystyle s_{i}}$ are 0) to ${\displaystyle \sum _{i\in I}{\sqrt {\left s_{i}\right }}}$ is a paranorm on ${\displaystyle X,}$ which satisfies ${\displaystyle p(sx)={\sqrt { s }}p(x)}$ for all ${\displaystyle x\in X}$ and scalars ${\dis$$놀이 스타일 s.}$
• The function ${\displaystyle p(x):= \sin(\pi x) +\min\{2, x \}}$is a paranorm on ${\displaystyle \mathbb {R} }$ that is not balanced but nevertheless equivalent to the usual norm on ${\displaystyle R.}$ Note that the function ${\display$$style$x\$mapsto \sin(\pi x)$ }은(는) 하위 추가 기능이다$$.^[10]
• Let ${\displaystyle X_{\mathbb {C} }}$ be a complex vector space and let ${\displaystyle X_{\mathbb {R} }}$ denote ${\displaystyle X_{\mathbb {C} }}$ considered as a vector space over ${\displaystyle \mathbb {R} .}$ Any paranorm on ${\displaystyle X_{\mathbb {C} }}$ is also$X{\displaystyle {}_{}}$ ${\displaystyle {R.}_{}}$ ${\displaystyle X_{\mathb {R}}}$에 나오는 편집광.

F-세미놈들

$X{\displaystyle }$ ${\displaystyle$ X$}$이(가) 실제 또는 복잡한 숫자에 대한 벡터 공간인$$ 경우 X ${\displaystyle$ X$}($$F{\displaystyle }$ ${\displaystyle$ $F$$}$은(는 Fréchet을 나타냄)의 F-seminorm은 다음과 같은 속성이 있는$$ 실제 값 $맵{\displaystyle }$ p :${\displaystyle }$X${\displaystyle }$→ ${\displaystyle R\mathbb{}}$ ${\displaystystyp$ {$RB}$이다$.$^[11]

1. 비음성: $p{\displaystyle }$≥ ${\displaystyle 0.}$ ${\displaystyle p\geq.}$
2. Subadditive: ${\displaystyle p}$( ${\displaystyle x}$+ y${\displaystyle }$)${\displaystyle \le }$ ${\displaystyle p}$( ${\displaystyle x}$)${\displaystyle }$+ p ( ${\displaystyle y}$ ) + p ( y ) 모든 x , ${\displaystyle y)X}$ ${\displaystyle p(x+y)\leq p(x)+p(y){\text{{}모든 }x,y\in$ X$\};;$
3. ${\displaystyle \mathrm{균형}}$ 조정: $p{\displaystyle }$( x${\displaystyle }$) ${\displaystyle \le }$ ${\displaystyle p}$( ${\displaystyle x}$) ${\displaystyle p$$($$ax)\leq$ p(x$)}$ 모든 $x x$ X {\$displaystyle x\in$ X$}$ 및$$$$ ${\displaystyle \le }$ 1 ${\displaystyle$a$\leq 1$을 충족하는$$ 모든$.$
   • This condition guarantees that each set of the form ${\displaystyle \{x\in X:p(x)\leq r\}}$ or ${\displaystyle \{x\in X:p(x)<r\}}$ for some ${\displaystyle r\geq 0}$ is balanced.
4. 매 $x{\displaystyle }$∈ ${\displaystyle X}$, ${\displaystyle x\in X,}$${\displaystyle }$(${\displaystyle \frac{\mathrm{1n}}{}}$ ${\displaystyle x}$)${\displaystyle }$→ ${\displaystyle 0}$ ${\displaystyle p\left ({\frac {1}{n}x\right)\to$ 0$}$에 대해 n${\displaystyle }$→ ${\displaystyle \mathrm{\infty }}$ ${\displaysty n\to }$
   • 시퀀스$({\displaystyle {\frac{\mathrm{1n}}{}}_{}^{}}$)${\displaystyle {}_{}^{}}$= ${\displaystyle 1_{}^{}}$ ${\displaystyle {\mathrm{}}_{}^{}}${\$displaystyle \left({\frac${$1}{n}\오른쪽)_{n=1}^{\flt}}}}}$은(는) 0으로 수렴하는 임의의 양의 시퀀스로 대체될$$ 수 있다.^[12]

F-세미놈은 F-norm이라고 불리는데, F-norm은 추가로 다음과 같은 조건을 F-norm이라고 부른다.

5. $총계{\displaystyle }$/${\displaystyle \mathrm{양수}}$ 한정자: p${\displaystyle }$( x${\displaystyle }$)${\displaystyle }$= ${\displaystyle 0}$ ${\displaystyle p(x)=0}$은 $x{\displaystyle }$= ${\displaystyle 0.}$ ${\displaystyle$ x$=0$을 의미한다$.$$}$

F-세미놈은 다음과 같은 경우에 모노톤이라고 불린다.

6. Monotone: ( x)p ( ) ( s ) p ( s ) ${\displaystyle p(rx)}을($를) 모두 0이 ${\displaystyle \mathrm{아닌}}$ 모든 x ${\displaystyle \in }$ X ${\displaystyle x\in X}$ 및$$ $모든{\displaystyle }$ $실제{\displaystyle }$ s${\displaystyt$ s}$및$ t$$ ${\\displaystystyt$$}$에 대해 "$p($s)".$}$^[12]

F-소형 공간

F-seminormed space (resp.F 표준 공간)은 ^[12]벡터 공간 X {\$displaystyle$ X$}과($resp) F-seminorm(resp)으로$$$$ 구성된 쌍$({\displaystyle X}$, ${\displaystyle p}$) ${\displaystyle(X,p)}$이다.F-norm) $X{\displaystyle }$. ${\displaystyle$ X$.}$의 p ${\displaystyle$p}

If ${\displaystyle (X,p)}$ and ${\displaystyle (Z,q)}$ are F-seminormed spaces then a map ${\displaystyle f:X\to Z}$ is called an isometric embedding^[12] if ${\displaystyle q(f(x)-f(y))=p(x,y){\text{ for all }}x,y\in X.}$

하나의 F-세미닌된 공간을 다른 공간에 내장하는 모든 등축은 위상학적 내장이지만, 그 반대는 일반적으로 사실이 아니다.^[12]

F-세미놈의 예

• F-세미놈의 모든 양성 스칼라 배수(resp.F-norm, 세미노름(resp)은 다시 F-seminorm(resp)이다.F-norm, 세미노름.
• 정확히 많은 F-세미놀이의 합계(resp.F-norms(F-norms)는 F-세미놈(resp)이다.F-norm).
• $p{\displaystyle }$ ${\displaystyle p}$ 및$$ $q{\displaystyle }$ ${\displaystyle q}$이(가) $X{\displaystyle }$ ${\\displaystyle X$}의 F-seminorms인 경우$$, 포인트우월성 $x{\displaystyle }$ $↦ supp${$($${\displaystystystylex\}$$$ $X{\displaystyle .}$ ${\displaystyle X.}$에 있는 어떤 비어 있지 않은 유한한 F-세미나미의 우월성도 마찬가지다.
• F-seminorm(resp)의 제한.벡터 서브스페이스에 대한 F-norm)은 F-seminorm(resp)이다.F-norm).^[9]
• $X{\displaystyle }$ ${\displaystyle X}$의 음이 아닌 실질 가치 함수는 볼록 F-세미나름인 경우 및 볼록 균형 G-세미나름인 경우, 동등하게만 세미노름이다$$.^[10]특히 모든 세미놈은 F-세미놈이다.
• ${\displaystyle {}^{}}$< 1,${\displaystyle$ $0$$<1,}$에 대해 지도 $f{\displaystyle }$${\$에 의해 정의됨
  $${\displaystyle [f\왼쪽(x_{1},\ldots,x_{n}\오른쪽)]^{p}=\좌측 x_{1}\우측 ^{p}+\cdots \좌측 x_{n}\우측 ^{p}}}$$

  
  표준이 아닌 F-표준이다.
• $L{\displaystyle }$: ${\displaystyle X}$→ ${\displaystyle Y}$ ${\displaystyle L:X\to$ Y$}$이(가) 선형 지도이고$$, $q{\displaystyle }${\$displaystyle$ y$}$이(가) $Y$ , ${\displaystyle Y,}$인 경우 $q{\displaystyle }$ ${\displaystyle \circ }$ ${\displaystyle L}$ ${\displaysty q$$\circircl}$은 $.$
• Let ${\displaystyle X_{\mathbb {C} }}$ be a complex vector space and let ${\displaystyle X_{\mathbb {R} }}$ denote ${\displaystyle X_{\mathbb {C} }}$ considered as a vector space over ${\displaystyle \mathbb {R} .}$ Any F-seminorm on ${\displaystyle X_{\mathbb {C} }}$ is al$그래서{\displaystyle {}_{}}$ X ${\displaystyle {R.}_{}}$ ${\displaystyle X_{\mathb {R}}}$에 F-seminorm.

F-세미놈의 특성

모든 F-seminorm은 편집증이고 모든 편집증은 어떤 F-seminorm과 동등하다.^[7]Every F-seminorm on a vector space ${\displaystyle X}$ is a value on ${\displaystyle X.}$ In particular, ${\displaystyle p(x)=0,}$ and ${\displaystyle p(x)=p(-x)}$ for all ${\displaystyle x\in X.}$

단일 F-세미놈에 의해 유도된 위상

F-세미놈 집단에 의해 유도된 위상

$L{\displaystyle \mathcal{}}$ ${\$이(가) 벡터 $공간{\displaystyle }$ X ${\displaystyle X}$ 및 유한$$ 부분 $집합{\displaystyle \mathcal{}}$ ${\displaystyle {F}\dubseteq {L}$과(r >0, {\$displaystystyle$ $r0}$에 대한 F-sminorms의 비어 있지 않은 집합이라고$$ 가정합시다.

세트{ , r: > , L, ${\$U_{{{F\mathcal}},r}일:~r>, 0,{{F\mathcal}}\subseteq{{나는\mathcal}},{{F\mathcal}}{\text{정형의}}\right\}}.}각[12]X{X\displaystyle}그것은 발신지 X{X\displaystyle}에 대한 벡터 토폴로지τ L.{\displaystyle \tau_{{\mathcal L}에 의해 표시된에}이웃 기초를 형성한 필터 근거지를 이루고 있다. UF${\displaystyle {,}_{}}$ ${\displaystyle r_{}}$ ${\$은(는) X의 균형잡히고 흡수적인 부분집합이다$.$ {\$displaystyle$ X$.}$ 이들$$^[12] 집합은 충족된다^[12]${\displaystyle .}$

• ${\displaystyle {\tau }_{}}$ ${\displaystyle {L\mathcal{}}_{}}$ ${\$ {$L}}$은(는) X ${\$$displaystyle X}$에서 각 $p{\displaystyle continuous\mathcal{}}$ L {\$displaystyle p\in {\mathcal {L}$을(를) 연속적으로 만드는$$ 가장 강력한 벡터 위상이다.^[12]
• ${\displaystyle {\tau }_{}}$ ${\displaystyle {L\mathcal{}}_{}}$ ${\$ 0이 아닌 모든 $x{\displaystyle }$∈ ${\displaystyle X,}$ ${\displaystyle$ $x\in {\mathcal {L}$에 p ) ${\displaysty$ p$(x)0$}과$$ 같은$$ $일부{\displaystyle }$ p${\displaystyle some\mathcal{}}$ L ${\\$$.$$}$^[12]
• If ${\displaystyle {\mathcal {F}}}$ is the set of all continuous F-seminorms on ${\displaystyle \left(X,\tau _{\mathcal {L}}\right)}$ then ${\displaystyle \tau _{\mathcal {L}}=\tau _{\mathcal {F}}.$$}$^[12]
• If ${\displaystyle {\mathcal {F}}}$ is the set of all pointwise suprema of non-empty finite subsets of ${\displaystyle {\mathcal {F}}}$ of ${\displaystyle {\mathcal {L}}}$ then ${\displaystyle {\mathcal {F}}}$ is a directed family of F-seminorms and ${\displaystyle \tau _{$$\mathcal{L}=\tau _{\mathcal {F}.$$}$^[12]

프레셰 조합

${\displaystyle {}_{}^{}}$ ${\displaystyle {\bullet }_{}}$=${\displaystyle {}_{}^{}}$( ${\displaystyle {p_{}}_{}^{}}$ ${\displaystyle {{}_{}}_{}^{}}$) i${\displaystyle {}_{}^{}}$= ${\displaystyle 1_{}^{}}$ ${\displaystyle {\mathrm{}}_{}^{}}$ ${\$이(가) 벡터 $공간{\displaystyle }$ X${\displaystyle }$.$${\$displaysty X.}$에 음이 아닌 하위 부가함수의 계열이라고 가정하자.

$p{\displaystyle {}_{}}$ ${\displaystyle {\bullet }_{}}$ ${\$의 프레셰 조합은^[8] 실제 값 맵으로 정의된다$$.

F-세미놈으로서

Assume that ${\displaystyle p_{\bullet }=\left(p_{i}\right)_{i=1}^{\infty }}$ is an increasing sequence of seminorms on ${\displaystyle X}$ and let ${\displaystyle p}$ be the Fréchet combination of ${\displaystyle p_{\bullet }.}$ Then ${\displaystyle p}$ is an F-se$X$ ${\displaystyle X}$에서 minorm p ${\displaystyle {\bullet }_{}}$ ${\$과(와) 동일한 국소 볼록 위상을 유도한다$$.^[13]

이후 p∙)(p나는)나는으로 범위{\displaystyle 나는}1∞{\displaystyle p_{\bullet}(p_{나는}\right)_{i=1}^{\infty}}, 발신지의 개방적인 지역 기반 형태{)∈ X:나는())<>r}{\displaystyle \left\{x\in X~:~p_{나는}())&lt의 모든 집합으로 이루어져 있고, r\right\}}o. 증가하고 있원얼음 알l 양의 정수 및 > ${\displaystyle r>0}$의 범위는 모든 양의 실제 숫자에 걸쳐 있다.

이 F-세미나름 $p{\displaystyle }$ ${\displaystyle p}$이(가$$) $유도$한$$X {\$displaystyle$X}의 번역 불변 가성계는 다음과$$ 같다.

이 측정기준은 1906년 프레셰트가 발표한 논문에서 포인트 와이즈 연산이 있는 실제적이고 복잡한 시퀀스의 공간에 대해 발견했다.^[14]

편집광으로서

각 $p{\displaystyle {}_{}}$ ${\displaystyle i_{}}$ ${\$가 편집증이라면$$ $p{\displaystyle }$ ${\displaystyle$ $p$$}$도 마찬가지며, 더욱이$$ $p{\displaystyle }$ ${\$$displaystyle$ $p}$은 편집증 $패밀리$ p ${\displaystyle {\bullet }_{}}${\$displaystyle p_{\bullet }}$과 같은 위상(위상)을$$유도한다.^[8]$이$는 X ${\displaystyle X}$의 다음과 같은 편집증에도 해당된다$.$

• ${\displaystyle q(x):=\inf _{}\left\{\sum _{i=1}^{np_{n}(x)+{\frac{1}{n}:{n}:{n}:n>0{\text}은(는) 정수 }}}}\right\}}.}$^[8]
• ${\displaystyle r(x):=\sum _{n=1}^{\influt \}\min \left\{1}{2^{n}},p_{n}(x)\right\}.}$^[8]

일반화

프리셰 조합은 경계 리메트리제이션 함수를 사용하여 일반화할 수 있다.

그것은 저가 산적(포지티브 R(의+t)≤ R(s)+R모든에, t≥ 0,{R(s+t)\leq R(s)+R(t)\displaystyle}(t){\displaystyle s,t\geq 0,}은 한정적 범위를 가지고 앉아 있는 한정적 remetrization)이 지속non-negativenon-decreasing 지도 R:[0, ∞)→ -LSB- 0, ∞){\displaystyle R:-LSB- 0,\infty)\to는 경우에는 0,\infty)}.isfies $R{\displaystyle (s}$ )${\displaystyle }$= ${\displaystyle 0}$ ${\displaystyle R$$($$s)=0}$ $s{\displaystyle }$= ${\displaystyle 0.}$ ${\displaystyle s=0$인 경우에만$$ 해당된다.$}$

Examples of bounded remetrization functions include ${\displaystyle \arctan t,}$ ${\displaystyle \tanh t,}$ ${\displaystyle t\mapsto \min\{t,1\},}$ and ${\displaystyle t\mapsto {\frac {t}{1+t}}.$}[15]X{X\displaystyle}과 R{R\displaystyle}에 만약 d{\displaystyle d}은 합치(resp. 계량)은 한정적 remetrization 기능 X{X\displaystyle}에 균일하게 삭제할 동등한 다음 R∘ d{\displaystyle R\circ d}은 경계 의거리(resp. 속박 미터).{\displaysty.이끌었다$.}$^[15]

Suppose that ${\displaystyle p_{\bullet }=\left(p_{i}\right)_{i=1}^{\infty }}$ is a family of non-negative F-seminorm on a vector space ${\displaystyle X,}$ ${\displaystyle R}$} is a bounded remetrization function, and ${\displaystyle r_{\bullet$$}}=\왼쪽(r_{i}\오른쪽)_{i=1}^{\ft{}}}}}$은 합이$$ 유한한 양의 실수의 순서다.그러면.

.{\displaystyle p_{\bullet}.}[16]이 얼마간의 네트 x∙에)}X에서,{X\displaystyle,}p(→ 0{\displaystyle p\left(x_{\bullet}\ri ∈{\displaystyle x_{\bullet}(x_{}\right)_{Aa\in}())는 속성을 갖는 한정적 F-seminorm는 한결같이 p∙에 해당합니다를 정의합니다.ght$)\to 0}$ if and only if ${\displaystyle p_{i}\left(x_{\bullet }\right)\to 0}$ for all ${\displaystyle i.}$^[16]${\displaystyle p}$ is an F-norm if and only if the ${\displaystyle p_{\bullet }}$ separate points on ${\displaystyle X.}$^[16]

특성화

(semi)규범에 의해 유도된 (pseudo)측정법 중

$d{\displaystyle }$ ${\displaystyle d}$이(가) 변환 불변성이고 절대적으로 동질적인 경우에만$$ 벡터 공간 $X{\displaystyle }$ ${\displaystyle X}$의 세미놈(resp. norm)에 의해 가성$$(resp. metric) $d}$이 $유도$되며, 이는 모든 스칼라 {\$displaystystyle$ s$}$과 ${\displaystyle \mathrm{모든}}$x ${\displaystyle ,}$ ${\x,$$${\$daysty$$.$$}$ ${\displaystyle 이}$ 경우$$ $p{\displaystyle }$( x${\displaystyle }$) :${\displaystyle d}$ (${\displaystyle \mathrm{x}}$, 0${\displaystyle }$) ${\displaystyle$ p$(x$):$d(x,0)$로 정의되는 함수는 세미노름(resp. norm)이고$$ p ${\displaystyle p}$에 의해 유도된 유사계(resp. metric$)$는 $d{\displaystyle }$. ${\displaystystyled$와 같다$$.$}$

가성계측 가능한 TV의 경우

$({\displaystyle }$ ${\displaystyle X}$, ${\displaystyle \tau }$) ${\displaystyle (X,\tau )}$이(가) 위상학적 벡터 공간(TV)인 경우(특히$display$ {\displaystyle $\tau}$이(가) 벡터 위상이라고 가정한다는$$ 유의)에는 다음이 동등하다.^[11]

1. ${\displaystyle X}$ ${\displaystyle X}$은(즉, 벡터 위상 $\tau {\displaystyle }$ ${\displaystyle \tau }$은(는) $X{\displaystyle }$ ${\displaystyle X}$의 유사 측정에 의해 유도된다$$.
2. ${\displaystyle X}$ ${\displaystyle X}$은(는) 출발지에 카운트 가능한 이웃 기지를 가지고 있다$$.
3. $X{\displaystyle }$ ${\displaystyle X$$}$의 $토폴로지$는${\displaystyle }$X$$. {\$displaystyle$ X}의 변환-증가 유사 측정값으로 유도된다$$.
4. $X{\displaystyle }$ ${\displaystyle X}$의 위상은 F-세미놈에 의해 유도된다$$.
5. $X{\displaystyle }$ ${\displaystyle X}$의 위상은 편집광에 의해 유도된다$$.

Metrizable TVs 중

$({\displaystyle }$ ${\displaystyle X}$, ${\displaystyle \tau }$) ${\displaystyle (X,\tau )}$이(가) TVS인$$ 경우, 다음과 같다.

1. ${\displaystyle X}$ ${\displaystyle X}$은(는) 미터링할 수 있다.
2. ${\displaystyle X}$ ${\displaystyle X}$은(는) 하우스도르프(Hausdorff)이며 가성측정가능하다.
3. ${\displaystyle X}$ ${\displaystyle X}$은(는) 하우스도르프(Hausdorff)이며$$ 원산지에는 셀 수 있는 근린기반이 있다.^[11][12]
4. $X{\displaystyle }$ ${\displaystyle X}$의 토폴로지는 $X{\displaystyle }$. ${\displaystyle X}$의 변환 내변성 메트릭에 의해 유도된다$$.
5. $X{\displaystyle }$ ${\displaystyle X}$의 위상은 F-orm에 의해 유도된다$$.^[11][12]
6. $X{\displaystyle }$ ${\displaystyle X}$의 위상은 단조 F-표준에 의해 유도된다$$.^[12]
7. $X{\displaystyle }$ ${\displaystyle X}$의 위상은 완전한 편집증에 의해 유도된다$$.

지역적으로 볼록한 가성계 TV의 경우

$({\displaystyle }$ ${\displaystyle X}$, ${\displaystyle \tau }$) ${\displaystyle (X,\tau )}$이(가) TVS일$$ 경우, 다음 사항은 동일하다.^[13]

1. ${\displaystyle X}$ ${\displaystyle X}$은(는) 국소적으로 볼록하고 유사하게 측정할 수 있다.
2. ${\displaystyle X}$ ${\displaystyle X}$은(는) 볼록 세트로 구성된 원점에 카운트 가능한 근린 기지를 가지고$$ 있다.
3. $X{\displaystyle }$ ${\displaystyle X}$의 위상은 (연속) 세미노름의 카운트 가능한 패밀리에 의해 유도된다$$.
4. The topology of ${\displaystyle X}$ is induced by a countable increasing sequence of (continuous) seminorms ${\displaystyle \left(p_{i}\right)_{i=1}^{\infty }}$ (increasing means that for all ${\displaystyle i,}$ ${\displaystyle p_{i}\geq p_{i+1}.$$}$
5. $X{\displaystyle }$ ${\displaystyle X}$의 토폴로지는 다음 형식의 F-세미나에 의해 유도된다$$.
   $${\displaystyle p(x)=\sum _{n=1}^{\infit }2^{-n}\nname {arctan}p_{n}(x)}$$

   
   ${\displaystyle {}_{}^{}}$서${\displaystyle {}_{}^{}}$(${\displaystyle {{}_{}}_{}^{}}$ ${\displaystyle {i_{}}_{}^{}}$ ${\displaystyle {{}_{}}_{}^{}}$ i${\displaystyle {}_{}^{}}$= ${\displaystyle 1_{}^{}}$ ${\displaystyle {\mathrm{}}_{}^{}}$ ${\$=1$}^{\nfty }}}}}$은 X . {\$displaystyle X.}$의 (연속) 세미orms이다.

인용구

$M{\displaystyle }$ ${\displaystyle M}$을(를) 위상학적 벡터 공간${\displaystyle (X}$,$τ$)의 벡터 $하위$ 공간이 되도록$$ 한다$.$$}$

• $X{\displaystyle }$ ${\displaystyle X}$이(가) 유사 측정 가능한 TVS인 경우 $X{\displaystyle /M.}$ ${\displaystyle$ X$/M$도 마찬가지다$.$$}$^[11]
• $X{\displaystyle }$ ${\displaystyle X}$이(가) 완전한 유사계측 가능 $TVS$이고 M$${\$displaystyle$M}이(가$)$ $X{\displaystyle }$ ${\displaystyle$ X$}$의 폐쇄 벡터 하위 공간이라면$$ X/${\displaystyle$ X/$M}이($가$$$$) 완료된다.^[11]
• $X{\displaystyle }$ ${\displaystyle X$$}$이(가) $Metrizeable{\displaystyle }$ TVS이고 M ${\displaystyle X}$의 폐쇄 벡터 하위 공간인$$ 경우 $X$ /${\displaystyle M}$ ${\displaystyle X/M}$을(를) Metrizeable로 한다$$.^[11]
• $p{\displaystyle }$ ${\displaystyle$ $p$$}$이(가) $X{\displaystyle }$, ${\$${\displaystyle displaystyle}$ X$,}$의 F-seminorm인 경우 지도$$ $P{\displaystyle }$: ${\displaystyle X}$/ M${\displaystyle }$→ ${\displaystyle R\mathbb{}}$ ${\$에 의해$$ 정의됨
  $${\displaystyle P(x+M):=\inf _{}\{p(x+m):m\in M\}}$$

  
  is an F-seminorm on ${\displaystyle X/M}$ that induces the usual quotient topology on ${\displaystyle X/M.}$^[11] If in addition ${\displaystyle p}$ is an F-norm on ${\displaystyle X}$ and if ${\displaystyle M}$ is a closed vector subspace of ${\displaystyle X}$ then ${\displaystyle P}$ is an F-norm on ${\displaystyle X.}$^[11]

Examples and sufficient conditions

• Every seminormed space ${\displaystyle (X,p)}$ is pseudometrizable with a canonical pseudometric given by ${\displaystyle d(x,y):=p(x-y)}$ for all ${\displaystyle x,y\in X.}$^[19].
• If ${\displaystyle (X,d)}$ is pseudometric TVS with a translation invariant pseudometric ${\displaystyle d,}$ then ${\displaystyle p(x):=d(x,0)}$ defines a paranorm.^[20] However, if ${\displaystyle d}$ is a translation invariant pseudometric on the vector space ${\displaystyle X}$ (without the addition condition that ${\displaystyle (X,d)}$ is pseudometric TVS), then ${\displaystyle d}$ need not be either an F-seminorm^[21] nor a paranorm.
• If a TVS has a bounded neighborhood of the origin then it is pseudometrizable; the converse is in general false.^[14]
• If a Hausdorff TVS has a bounded neighborhood of the origin then it is metrizable.^[14]
• Suppose ${\displaystyle X}$ is either a DF-space or an LM-space. If ${\displaystyle X}$ is a sequential space then it is either metrizable or else a Montel DF-space.

If ${\displaystyle X}$ is Hausdorff locally convex TVS then ${\displaystyle X}$ with the strong topology, ${\displaystyle \left(X,b\left(X,X^{\prime }\right)\right),}$ is metrizable if and only if there exists a countable set ${\displaystyle {\mathcal {B}}}$ of bounded subsets of ${\displaystyle X}$ such that every bounded subset of ${\displaystyle X}$ is contained in some element of ${\displaystyle {\mathcal {B}}.}$^[22]

The strong dual space ${\displaystyle X_{b}^{\prime }}$ of a metrizable locally convex space (such as a Fréchet space^[23]) ${\displaystyle X}$ is a DF-space.^[24] The strong dual of a DF-space is a Fréchet space.^[25] The strong dual of a reflexive Fréchet space is a bornological space.^[24] The strong bidual (that is, the strong dual space of the strong dual space) of a metrizable locally convex space is a Fréchet space.^[26] If ${\displaystyle X}$ is a metrizable locally convex space then its strong dual ${\displaystyle X_{b}^{\prime }}$ has one of the following properties, if and only if it has all of these properties: (1) bornological, (2) infrabarreled, (3) barreled.^[26]

Normability

A topological vector space is seminormable if and only if it has a convex bounded neighborhood of the origin. Moreover, a TVS is normable if and only if it is Hausdorff and seminormable.^[14] Every metrizable TVS on a finite-dimensional vector space is a normable locally convex complete TVS, being TVS-isomorphic to Euclidean space. Consequently, any metrizable TVS that is not normable must be infinite dimensional.

If ${\displaystyle M}$ is a metrizable locally convex TVS that possess a countable fundamental system of bounded sets, then ${\displaystyle M}$ is normable.^[27]

If ${\displaystyle X}$ is a Hausdorff locally convex space then the following are equivalent:

1. ${\displaystyle X}$ is normable.
2. ${\displaystyle X}$ has a (von Neumann) bounded neighborhood of the origin.
3. the strong dual space ${\displaystyle X_{b}^{\prime }}$ of ${\displaystyle X}$ is normable.^[28]

and if this locally convex space ${\displaystyle X}$ is also metrizable, then the following may be appended to this list:

4. the strong dual space of ${\displaystyle X}$ is metrizable.^[28]
5. the strong dual space of ${\displaystyle X}$ is a Fréchet–Urysohn locally convex space.^[23]

In particular, if a metrizable locally convex space ${\displaystyle X}$ (such as a Fréchet space) is not normable then its strong dual space ${\displaystyle X_{b}^{\prime }}$ is not a Fréchet–Urysohn space and consequently, this complete Hausdorff locally convex space ${\displaystyle X_{b}^{\prime }}$ is also neither metrizable nor normable.

Another consequence of this is that if ${\displaystyle X}$ is a reflexive locally convex TVS whose strong dual ${\displaystyle X_{b}^{\prime }}$ is metrizable then ${\displaystyle X_{b}^{\prime }}$ is necessarily a reflexive Fréchet space, ${\displaystyle X}$ is a DF-space, both ${\displaystyle X}$ and ${\displaystyle X_{b}^{\prime }}$ are necessarily complete Hausdorff ultrabornological distinguished webbed spaces, and moreover, ${\displaystyle X_{b}^{\prime }}$ is normable if and only if ${\displaystyle X}$ is normable if and only if ${\displaystyle X}$ is Fréchet–Urysohn if and only if ${\displaystyle X}$ is metrizable. In particular, such a space ${\displaystyle X}$ is either a Banach space or else it is not even a Fréchet–Urysohn space.

Metrically bounded sets and bounded sets

Suppose that ${\displaystyle (X,d)}$ is a pseudometric space and ${\displaystyle B\subseteq X.}$ The set ${\displaystyle B}$ is metrically bounded or ${\displaystyle d}$-bounded if there exists a real number ${\displaystyle R>0}$ such that ${\displaystyle d(x,y)\leq R}$ for all ${\displaystyle x,y\in B}$; the smallest such ${\displaystyle R}$ is then called the diameter or ${\displaystyle d}$-diameter of ${\displaystyle B.}$^[14] If ${\displaystyle B}$ is bounded in a pseudometrizable TVS ${\displaystyle X}$ then it is metrically bounded; the converse is in general false but it is true for locally convex metrizable TVSs.^[14]

Properties of pseudometrizable TVS

• Every metrizable locally convex TVS is a quasibarrelled space,^[30] bornological space, and a Mackey space.
• Every complete pseudometrizable TVS is a barrelled space and a Baire space (and hence non-meager).^[31] However, there exist metrizable Baire spaces that are not complete.^[31]
• If ${\displaystyle X}$ is a metrizable locally convex space, then the strong dual of ${\displaystyle X}$ is bornological if and only if it is barreled, if and only if it is infrabarreled.^[26]
• If ${\displaystyle X}$ is a complete pseudometrizable TVS and ${\displaystyle M}$ is a closed vector subspace of ${\displaystyle X,}$ then ${\displaystyle X/M}$ is complete.^[11]
• The strong dual of a locally convex metrizable TVS is a webbed space.^[32]
• If ${\displaystyle (X,\tau )}$ and ${\displaystyle (X,\nu )}$ are complete metrizable TVSs (i.e. F-spaces) and if ${\displaystyle \nu }$ is coarser than ${\displaystyle \tau }$ then ${\displaystyle \tau =\nu }$;^[33] this is no longer guaranteed to be true if any one of these metrizable TVSs is not complete.^[34] Said differently, if ${\displaystyle (X,\tau )}$ and ${\displaystyle (X,\nu )}$ are both F-spaces but with different topologies, then neither one of ${\displaystyle \tau }$ and ${\displaystyle \nu }$ contains the other as a subset. One particular consequence of this is, for example, that if ${\displaystyle (X,p)}$ is a Banach space and ${\displaystyle (X,q)}$ is some other normed space whose norm-induced topology is finer than (or alternatively, is coarser than) that of ${\displaystyle (X,p)}$ (i.e. if ${\displaystyle p\leq Cq}$ or if ${\displaystyle q\leq Cp}$ for some constant ${\displaystyle C>0}$), then the only way that ${\displaystyle (X,q)}$ can be a Banach space (i.e. also be complete) is if these two norms ${\displaystyle p}$ and ${\displaystyle q}$ are equivalent; if they are not equivalent, then ${\displaystyle (X,q)}$ can not be a Banach space. As another consequence, if ${\displaystyle (X,p)}$ is a Banach space and ${\displaystyle (X,\nu )}$ is a Fréchet space, then the map ${\displaystyle p:(X,\nu )\to \mathbb {R} }$ is continuous if and only if the Fréchet space ${\displaystyle (X,\nu )}$ is the TVS ${\displaystyle (X,p)}$ (here, the Banach space ${\displaystyle (X,p)}$ is being considered as a TVS, which means that its norm is "forgetten" but its topology is remembered).
• A metrizable locally convex space is normable if and only if its strong dual space is a Fréchet–Urysohn locally convex space.^[23]
• Any product of complete metrizable TVSs is a Baire space.^[31]
• A product of metrizable TVSs is metrizable if and only if it all but at most countably many of these TVSs have dimension ${\displaystyle 0.}$^[35]
• A product of pseudometrizable TVSs is pseudometrizable if and only if it all but at most countably many of these TVSs have the trivial topology.
• Every complete pseudometrizable TVS is a barrelled space and a Baire space (and thus non-meager).^[31]
• The dimension of a complete metrizable TVS is either finite or uncountable.^[35]

Completeness

Every topological vector space (and more generally, a topological group) has a canonical uniform structure, induced by its topology, which allows the notions of completeness and uniform continuity to be applied to it. If ${\displaystyle X}$ is a metrizable TVS and ${\displaystyle d}$ is a metric that defines ${\displaystyle X}$'s topology, then its possible that ${\displaystyle X}$ is complete as a TVS (i.e. relative to its uniformity) but the metric ${\displaystyle d}$ is not a complete metric (such metrics exist even for ${\displaystyle X=\mathbb {R} }$). Thus, if ${\displaystyle X}$ is a TVS whose topology is induced by a pseudometric ${\displaystyle d,}$ then the notion of completeness of ${\displaystyle X}$ (as a TVS) and the notion of completeness of the pseudometric space ${\displaystyle (X,d)}$ are not always equivalent. The next theorem gives a condition for when they are equivalent:

If ${\displaystyle M}$ is a closed vector subspace of a complete pseudometrizable TVS ${\displaystyle X,}$ then the quotient space ${\displaystyle X/M}$ is complete.^[40] If ${\displaystyle M}$ is a complete vector subspace of a metrizable TVS ${\displaystyle X}$ and if the quotient space ${\displaystyle X/M}$ is complete then so is ${\displaystyle X.}$^[40] If ${\displaystyle X}$ is not complete then ${\displaystyle M:=X,}$ but not complete, vector subspace of ${\displaystyle X.}$

A Baire separable topological group is metrizable if and only if it is cosmic.^[23]

Subsets and subsequences

• Let ${\displaystyle M}$ be a separable locally convex metrizable topological vector space and let ${\displaystyle C}$ be its completion. If ${\displaystyle S}$ is a bounded subset of ${\displaystyle C}$ then there exists a bounded subset ${\displaystyle R}$ of ${\displaystyle X}$ such that ${\displaystyle S\subseteq \operatorname {cl} _{C}R.}$^[41]
• Every totally bounded subset of a locally convex metrizable TVS ${\displaystyle X}$ is contained in the closed convex balanced hull of some sequence in ${\displaystyle X}$ that converges to ${\displaystyle 0.}$
• In a pseudometrizable TVS, every bornivore is a neighborhood of the origin.^[42]
• If ${\displaystyle d}$ is a translation invariant metric on a vector space ${\displaystyle X,}$ then ${\displaystyle d(nx,0)\leq nd(x,0)}$ for all ${\displaystyle x\in X}$ and every positive integer ${\displaystyle n.}$^[43]
• If ${\displaystyle \left(x_{i}\right)_{i=1}^{\infty }}$ is a null sequence (that is, it converges to the origin) in a metrizable TVS then there exists a sequence ${\displaystyle \left(r_{i}\right)_{i=1}^{\infty }}$ of positive real numbers diverging to ${\displaystyle \infty }$ such that ${\displaystyle \left(r_{i}x_{i}\right)_{i=1}^{\infty }\to 0.}$^[43]
• A subset of a complete metric space is closed if and only if it is complete. If a space ${\displaystyle X}$ is not complete, then ${\displaystyle X}$ is a closed subset of ${\displaystyle X}$ that is not complete.
• If ${\displaystyle X}$ is a metrizable locally convex TVS then for every bounded subset ${\displaystyle B}$ of ${\displaystyle X,}$ there exists a bounded disk ${\displaystyle D}$ in ${\displaystyle X}$ such that ${\displaystyle B\subseteq X_{D},}$ and both ${\displaystyle X}$ and the auxiliary normed space ${\displaystyle X_{D}}$ induce the same subspace topology on ${\displaystyle B.}$^[44]

Linear maps

If ${\displaystyle X}$ is a pseudometrizable TVS and ${\displaystyle A}$ maps bounded subsets of ${\displaystyle X}$ to bounded subsets of ${\displaystyle Y,}$ then ${\displaystyle A}$ is continuous.^[14] Discontinuous linear functionals exist on any infinite-dimensional pseudometrizable TVS.^[46] Thus, a pseudometrizable TVS is finite-dimensional if and only if its continuous dual space is equal to its algebraic dual space.^[46]

If ${\displaystyle F:X\to Y}$ is a linear map between TVSs and ${\displaystyle X}$ is metrizable then the following are equivalent:

1. ${\displaystyle F}$ is continuous;
2. ${\displaystyle F}$ is a (locally) bounded map (that is, ${\displaystyle F}$ maps (von Neumann) bounded subsets of ${\displaystyle X}$ to bounded subsets of ${\displaystyle Y}$);^[12]
3. ${\displaystyle F}$ is sequentially continuous;^[12]
4. the image under ${\displaystyle F}$ of every null sequence in ${\displaystyle X}$ is a bounded set^[12] where by definition, a null sequence is a sequence that converges to the origin.
5. ${\displaystyle F}$ maps null sequences to null sequences;

Open and almost open maps

Theorem: If ${\displaystyle X}$ is a complete pseudometrizable TVS, ${\displaystyle Y}$ is a Hausdorff TVS, and ${\displaystyle T:X\to Y}$ is a closed and almost open linear surjection, then ${\displaystyle T}$ is an open map.^[47]

Theorem: If ${\displaystyle T:X\to Y}$ is a surjective linear operator from a locally convex space ${\displaystyle X}$ onto a barrelled space ${\displaystyle Y}$ (e.g. every complete pseudometrizable space is barrelled) then ${\displaystyle T}$ is almost open.^[47]

Theorem: If ${\displaystyle T:X\to Y}$ is a surjective linear operator from a TVS ${\displaystyle X}$ onto a Baire space ${\displaystyle Y}$ then ${\displaystyle T}$ is almost open.^[47]

Theorem: Suppose ${\displaystyle T:X\to Y}$ is a continuous linear operator from a complete pseudometrizable TVS ${\displaystyle X}$ into a Hausdorff TVS ${\displaystyle Y.}$ If the image of ${\displaystyle T}$ is non-meager in ${\displaystyle Y}$ then ${\displaystyle T:X\to Y}$ is a surjective open map and ${\displaystyle Y}$ is a complete metrizable space.^[47]

Hahn-Banach extension property

A vector subspace ${\displaystyle M}$ of a TVS ${\displaystyle X}$ has the extension property if any continuous linear functional on ${\displaystyle M}$ can be extended to a continuous linear functional on ${\displaystyle X.}$^[22] Say that a TVS ${\displaystyle X}$ has the Hahn-Banach extension property (HBEP) if every vector subspace of ${\displaystyle X}$ has the extension property.^[22]

The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP. For complete metrizable TVSs there is a converse:

If a vector space ${\displaystyle X}$ has uncountable dimension and if we endow it with the finest vector topology then this is a TVS with the HBEP that is neither locally convex or metrizable.^[22]

See also

• Asymmetric norm – Generalization of the concept of a norm
• Complete metric space – Metric geometry
• Complete topological vector space – A TVS where points that get progressively closer to each other will always converge to a point
• Closed graph theorem (functional analysis) – Theorems for deducing continuity
• Equivalence of metrics
• F-space – Topological vector space with a complete translation-invariant metric
• Fréchet space – A locally convex topological vector space that is also a complete metric space
• K-space (functional analysis)
• Locally convex topological vector space – A vector space with a topology defined by convex open sets
• Metric space – Mathematical space with a notion of distance
• Open mapping theorem (functional analysis) – Condition for a linear operator to be open
• Pseudometric space – Generalization of metric spaces in mathematics
• Relation of norms and metrics
• Seminorm
• Sublinear function
• Topological vector space – Vector space with a notion of nearness
• Uniform space – Topological space with a notion of uniform properties
• Ursescu theorem – Generalization of closed graph, open mapping, and uniform boundedness theorem

Notes

References

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