Misplaced Pages

#426573

64-433: A pseudometric on a set X {\displaystyle X} is a map d : X × X → R {\displaystyle d:X\times X\rightarrow \mathbb {R} } satisfying the following properties: A pseudometric is called a metric if it satisfies: Ultrapseudometric A pseudometric d {\displaystyle d} on X {\displaystyle X}

128-455: A G-norm if it satisfies the additional condition: If p {\displaystyle p} is a value on a vector space X {\displaystyle X} then: Theorem  —  Suppose that X {\displaystyle X} is an additive commutative group. If d {\displaystyle d} is a translation invariant pseudometric on X {\displaystyle X} then

192-480: A topological vector space ). Every topological vector space (TVS) X {\displaystyle X} is an additive commutative topological group but not all group topologies on X {\displaystyle X} are vector topologies. This is because despite it making addition and negation continuous, a group topology on a vector space X {\displaystyle X} may fail to make scalar multiplication continuous. For instance,

256-421: A Hausdorff TVS with a countable basis of neighborhoods is metrizable. The following theorem is true more generally for commutative additive topological groups . Theorem  —  Let U ∙ = ( U i ) i = 0 ∞ {\displaystyle U_{\bullet }=\left(U_{i}\right)_{i=0}^{\infty }} be a collection of subsets of

320-981: A filter base on X {\displaystyle X} that also forms a neighborhood basis at the origin for a vector topology on X {\displaystyle X} denoted by τ L . {\displaystyle \tau _{\mathcal {L}}.} Each U F , r {\displaystyle U_{{\mathcal {F}},r}} is a balanced and absorbing subset of X . {\displaystyle X.} These sets satisfy U F , r / 2 + U F , r / 2 ⊆ U F , r . {\displaystyle U_{{\mathcal {F}},r/2}+U_{{\mathcal {F}},r/2}\subseteq U_{{\mathcal {F}},r}.} Suppose that p ∙ = ( p i ) i = 1 ∞ {\displaystyle p_{\bullet }=\left(p_{i}\right)_{i=1}^{\infty }}

384-2642: A finite sequence of non-negative integers and use the notation: ∑ 2 − n ∙ := 2 − n 1 + ⋯ + 2 − n k  and  ∑ U n ∙ := U n 1 + ⋯ + U n k . {\displaystyle \sum 2^{-n_{\bullet }}:=2^{-n_{1}}+\cdots +2^{-n_{k}}\quad {\text{ and }}\quad \sum U_{n_{\bullet }}:=U_{n_{1}}+\cdots +U_{n_{k}}.} For any integers n ≥ 0 {\displaystyle n\geq 0} and d > 2 , {\displaystyle d>2,} U n ⊇ U n + 1 + U n + 1 ⊇ U n + 1 + U n + 2 + U n + 2 ⊇ U n + 1 + U n + 2 + ⋯ + U n + d + U n + d + 1 + U n + d + 1 . {\displaystyle U_{n}\supseteq U_{n+1}+U_{n+1}\supseteq U_{n+1}+U_{n+2}+U_{n+2}\supseteq U_{n+1}+U_{n+2}+\cdots +U_{n+d}+U_{n+d+1}+U_{n+d+1}.} From this it follows that if n ∙ = ( n 1 , … , n k ) {\displaystyle n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)} consists of distinct positive integers then ∑ U n ∙ ⊆ U − 1 + min ( n ∙ ) . {\displaystyle \sum U_{n_{\bullet }}\subseteq U_{-1+\min \left(n_{\bullet }\right)}.} It will now be shown by induction on k {\displaystyle k} that if n ∙ = ( n 1 , … , n k ) {\displaystyle n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)} consists of non-negative integers such that ∑ 2 − n ∙ ≤ 2 − M {\displaystyle \sum 2^{-n_{\bullet }}\leq 2^{-M}} for some integer M ≥ 0 {\displaystyle M\geq 0} then ∑ U n ∙ ⊆ U M . {\displaystyle \sum U_{n_{\bullet }}\subseteq U_{M}.} This

448-656: A homogeneous, translation-invariant pseudometric induces a seminorm. Pseudometrics also arise in the theory of hyperbolic complex manifolds : see Kobayashi metric . Every measure space ( Ω , A , μ ) {\displaystyle (\Omega ,{\mathcal {A}},\mu )} can be viewed as a complete pseudometric space by defining d ( A , B ) := μ ( A △ B ) {\displaystyle d(A,B):=\mu (A\vartriangle B)} for all A , B ∈ A , {\displaystyle A,B\in {\mathcal {A}},} where

512-623: A metric space, points in a pseudometric space need not be distinguishable ; that is, one may have d ( x , y ) = 0 {\displaystyle d(x,y)=0} for distinct values x ≠ y . {\displaystyle x\neq y.} Any metric space is a pseudometric space. Pseudometrics arise naturally in functional analysis . Consider the space F ( X ) {\displaystyle {\mathcal {F}}(X)} of real-valued functions f : X → R {\displaystyle f:X\to \mathbb {R} } together with

576-457: A pseudometric on X 1 . If d 2 is a metric and f is injective , then d 1 is a metric. The pseudometric topology is the topology generated by the open balls B r ( p ) = { x ∈ X : d ( p , x ) < r } , {\displaystyle B_{r}(p)=\{x\in X:d(p,x)<r\},} which form a basis for

640-584: A special point x 0 ∈ X . {\displaystyle x_{0}\in X.} This point then induces a pseudometric on the space of functions, given by d ( f , g ) = | f ( x 0 ) − g ( x 0 ) | {\displaystyle d(f,g)=\left|f(x_{0})-g(x_{0})\right|} for f , g ∈ F ( X ) {\displaystyle f,g\in {\mathcal {F}}(X)} A seminorm p {\displaystyle p} induces

704-458: A vector space X {\displaystyle X} and an F -seminorm (resp. F -norm) p {\displaystyle p} on X . {\displaystyle X.} If ( X , p ) {\displaystyle (X,p)} and ( Z , q ) {\displaystyle (Z,q)} are F -seminormed spaces then a map f : X → Z {\displaystyle f:X\to Z}

SECTION 10

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768-444: A vector space X . {\displaystyle X.} Then the map d : X × X → R {\displaystyle d:X\times X\to \mathbb {R} } defined by d ( x , y ) := p ( x − y ) {\displaystyle d(x,y):=p(x-y)} is a translation invariant pseudometric on X {\displaystyle X} that defines

832-474: A vector space is called additive if for every N ∈ N , {\displaystyle N\in {\mathcal {N}},} there exists some U ∈ N {\displaystyle U\in {\mathcal {N}}} such that U + U ⊆ N . {\displaystyle U+U\subseteq N.} Continuity of addition at 0  —  If ( X , + ) {\displaystyle (X,+)}

896-2157: A vector space such that 0 ∈ U i {\displaystyle 0\in U_{i}} and U i + 1 + U i + 1 ⊆ U i {\displaystyle U_{i+1}+U_{i+1}\subseteq U_{i}} for all i ≥ 0. {\displaystyle i\geq 0.} For all u ∈ U 0 , {\displaystyle u\in U_{0},} let S ( u ) := { n ∙ = ( n 1 , … , n k )   :   k ≥ 1 , n i ≥ 0  for all  i ,  and  u ∈ U n 1 + ⋯ + U n k } . {\displaystyle \mathbb {S} (u):=\left\{n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)~:~k\geq 1,n_{i}\geq 0{\text{ for all }}i,{\text{ and }}u\in U_{n_{1}}+\cdots +U_{n_{k}}\right\}.} Define f : X → [ 0 , 1 ] {\displaystyle f:X\to [0,1]} by f ( x ) = 1 {\displaystyle f(x)=1} if x ∉ U 0 {\displaystyle x\not \in U_{0}} and otherwise let f ( x ) := inf { 2 − n 1 + ⋯ 2 − n k   :   n ∙ = ( n 1 , … , n k ) ∈ S ( x ) } . {\displaystyle f(x):=\inf _{}\left\{2^{-n_{1}}+\cdots 2^{-n_{k}}~:~n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)\in \mathbb {S} (x)\right\}.} Then f {\displaystyle f}

960-775: A vector topology τ {\displaystyle \tau } on X . {\displaystyle X.} If p {\displaystyle p} is an F -norm then d {\displaystyle d} is a metric. When X {\displaystyle X} is endowed with this topology then p {\displaystyle p} is a continuous map on X . {\displaystyle X.} The balanced sets { x ∈ X   :   p ( x ) ≤ r } , {\displaystyle \{x\in X~:~p(x)\leq r\},} as r {\displaystyle r} ranges over

1024-1294: Is subadditive (meaning f ( x + y ) ≤ f ( x ) + f ( y )  for all  x , y ∈ X {\displaystyle f(x+y)\leq f(x)+f(y){\text{ for all }}x,y\in X} ) and f = 0 {\displaystyle f=0} on ⋂ i ≥ 0 U i , {\displaystyle \bigcap _{i\geq 0}U_{i},} so in particular f ( 0 ) = 0. {\displaystyle f(0)=0.} If all U i {\displaystyle U_{i}} are symmetric sets then f ( − x ) = f ( x ) {\displaystyle f(-x)=f(x)} and if all U i {\displaystyle U_{i}} are balanced then f ( s x ) ≤ f ( x ) {\displaystyle f(sx)\leq f(x)} for all scalars s {\displaystyle s} such that | s | ≤ 1 {\displaystyle |s|\leq 1} and all x ∈ X . {\displaystyle x\in X.} If X {\displaystyle X}

1088-448: Is Hausdorff then the word "pseudometric" in the above statement may be replaced by the word "metric." A commutative topological group is metrizable if and only if it is Hausdorff and pseudometrizable. Let X {\displaystyle X} be a non-trivial (i.e. X ≠ { 0 } {\displaystyle X\neq \{0\}} ) real or complex vector space and let d {\displaystyle d} be

1152-406: Is a group (as all vector spaces are), τ {\displaystyle \tau } is a topology on X , {\displaystyle X,} and X × X {\displaystyle X\times X} is endowed with the product topology , then the addition map X × X → X {\displaystyle X\times X\to X} (i.e.

1216-424: Is a metric space , every seminormed space is a pseudometric space. Because of this analogy, the term semimetric space (which has a different meaning in topology ) is sometimes used as a synonym, especially in functional analysis . When a topology is generated using a family of pseudometrics, the space is called a gauge space . A pseudometric space ( X , d ) {\displaystyle (X,d)}

1280-400: Is a family of non-negative subadditive functions on a vector space X . {\displaystyle X.} Pseudometric space In mathematics , a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa in 1934. In the same way as every normed space

1344-412: Is a metric on X ∗ {\displaystyle X^{*}} and ( X ∗ , d ∗ ) {\displaystyle (X^{*},d^{*})} is a well-defined metric space, called the metric space induced by the pseudometric space ( X , d ) {\displaystyle (X,d)} . The metric identification preserves

SECTION 20

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1408-535: Is a pseudometric on a set X {\displaystyle X} then collection of open balls : B r ( z ) := { x ∈ X : d ( x , z ) < r } {\displaystyle B_{r}(z):=\{x\in X:d(x,z)<r\}} as z {\displaystyle z} ranges over X {\displaystyle X} and r > 0 {\displaystyle r>0} ranges over

1472-430: Is a real-valued map p : X → R {\displaystyle p:X\to \mathbb {R} } with the following four properties: An F -seminorm is called an F -norm if in addition it satisfies: An F -seminorm is called monotone if it satisfies: An F -seminormed space (resp. F -normed space ) is a pair ( X , p ) {\displaystyle (X,p)} consisting of

1536-438: Is a set X {\displaystyle X} together with a non-negative real-valued function d : X × X ⟶ R ≥ 0 , {\displaystyle d:X\times X\longrightarrow \mathbb {R} _{\geq 0},} called a pseudometric , such that for every x , y , z ∈ X , {\displaystyle x,y,z\in X,} Unlike

1600-406: Is a topological vector space and if all U i {\displaystyle U_{i}} are neighborhoods of the origin then f {\displaystyle f} is continuous, where if in addition X {\displaystyle X} is Hausdorff and U ∙ {\displaystyle U_{\bullet }} forms a basis of balanced neighborhoods of

1664-504: Is a translation-invariant pseudometric on X {\displaystyle X} and the value associated with d {\displaystyle d} is just p . {\displaystyle p.} Theorem  —  If ( X , τ ) {\displaystyle (X,\tau )} is an additive commutative topological group then the following are equivalent: If ( X , τ ) {\displaystyle (X,\tau )}

1728-560: Is a translation-invariant pseudometric on X {\displaystyle X} that defines a vector topology on X . {\displaystyle X.} If p {\displaystyle p} is a paranorm on a vector space X {\displaystyle X} then: If X {\displaystyle X} is a vector space over the real or complex numbers then an F -seminorm on X {\displaystyle X} (the F {\displaystyle F} stands for Fréchet )

1792-890: Is a vector space over the real or complex numbers then a paranorm on X {\displaystyle X} is a G-seminorm (defined above) p : X → R {\displaystyle p:X\rightarrow \mathbb {R} } on X {\displaystyle X} that satisfies any of the following additional conditions, each of which begins with "for all sequences x ∙ = ( x i ) i = 1 ∞ {\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }} in X {\displaystyle X} and all convergent sequences of scalars s ∙ = ( s i ) i = 1 ∞ {\displaystyle s_{\bullet }=\left(s_{i}\right)_{i=1}^{\infty }} ": A paranorm

1856-493: Is an additive group endowed with a topology, called a group topology , under which addition and negation become continuous operators. A topology τ {\displaystyle \tau } on a real or complex vector space X {\displaystyle X} is called a vector topology or a TVS topology if it makes the operations of vector addition and scalar multiplication continuous (that is, if it makes X {\displaystyle X} into

1920-772: Is an exercise. If all U i {\displaystyle U_{i}} are symmetric then x ∈ ∑ U n ∙ {\displaystyle x\in \sum U_{n_{\bullet }}} if and only if − x ∈ ∑ U n ∙ {\displaystyle -x\in \sum U_{n_{\bullet }}} from which it follows that f ( − x ) ≤ f ( x ) {\displaystyle f(-x)\leq f(x)} and f ( − x ) ≥ f ( x ) . {\displaystyle f(-x)\geq f(x).} If all U i {\displaystyle U_{i}} are balanced then

1984-423: Is called pseudometrizable (resp. metrizable , ultrapseudometrizable ) if there exists a pseudometric (resp. metric, ultrapseudometric) d {\displaystyle d} on X {\displaystyle X} such that τ {\displaystyle \tau } is equal to the topology induced by d . {\displaystyle d.} An additive topological group

Metrizable topological vector space - Misplaced Pages Continue

2048-449: Is called total if in addition it satisfies: If p {\displaystyle p} is a paranorm on a vector space X {\displaystyle X} then the map d : X × X → R {\displaystyle d:X\times X\rightarrow \mathbb {R} } defined by d ( x , y ) := p ( x − y ) {\displaystyle d(x,y):=p(x-y)}

2112-455: Is called a ultrapseudometric or a strong pseudometric if it satisfies: Pseudometric space A pseudometric space is a pair ( X , d ) {\displaystyle (X,d)} consisting of a set X {\displaystyle X} and a pseudometric d {\displaystyle d} on X {\displaystyle X} such that X {\displaystyle X} 's topology

2176-408: Is called an isometric embedding if q ( f ( x ) − f ( y ) ) = p ( x , y )  for all  x , y ∈ X . {\displaystyle q(f(x)-f(y))=p(x,y){\text{ for all }}x,y\in X.} Every isometric embedding of one F -seminormed space into another is a topological embedding , but the converse

2240-423: Is clearly true for k = 1 {\displaystyle k=1} and k = 2 {\displaystyle k=2} so assume that k > 2 , {\displaystyle k>2,} which implies that all n i {\displaystyle n_{i}} are positive. If all n i {\displaystyle n_{i}} are distinct then this step

2304-702: Is done, and otherwise pick distinct indices i < j {\displaystyle i<j} such that n i = n j {\displaystyle n_{i}=n_{j}} and construct m ∙ = ( m 1 , … , m k − 1 ) {\displaystyle m_{\bullet }=\left(m_{1},\ldots ,m_{k-1}\right)} from n ∙ {\displaystyle n_{\bullet }} by replacing each n i {\displaystyle n_{i}} with n i − 1 {\displaystyle n_{i}-1} and deleting

2368-426: Is identical to the topology on X {\displaystyle X} induced by d . {\displaystyle d.} We call a pseudometric space ( X , d ) {\displaystyle (X,d)} a metric space (resp. ultrapseudometric space ) when d {\displaystyle d} is a metric (resp. ultrapseudometric). If d {\displaystyle d}

2432-665: Is not true in general. Every F -seminorm is a paranorm and every paranorm is equivalent to some F -seminorm. Every F -seminorm on a vector space X {\displaystyle X} is a value on X . {\displaystyle X.} In particular, p ( x ) = 0 , {\displaystyle p(x)=0,} and p ( x ) = p ( − x ) {\displaystyle p(x)=p(-x)} for all x ∈ X . {\displaystyle x\in X.} Theorem  —  Let p {\displaystyle p} be an F -seminorm on

2496-536: Is subadditive, it suffices to prove that f ( x + y ) ≤ f ( x ) + f ( y ) {\displaystyle f(x+y)\leq f(x)+f(y)} when x , y ∈ X {\displaystyle x,y\in X} are such that f ( x ) + f ( y ) < 1 , {\displaystyle f(x)+f(y)<1,} which implies that x , y ∈ U 0 . {\displaystyle x,y\in U_{0}.} This

2560-403: Is that scalar multiplication isn't continuous on ( X , τ ) . {\displaystyle (X,\tau ).} This example shows that a translation-invariant (pseudo)metric is not enough to guarantee a vector topology, which leads us to define paranorms and F -seminorms. A collection N {\displaystyle {\mathcal {N}}} of subsets of

2624-422: Is the discrete topology , which makes ( X , τ ) {\displaystyle (X,\tau )} into a commutative topological group under addition but does not form a vector topology on X {\displaystyle X} because ( X , τ ) {\displaystyle (X,\tau )} is disconnected but every vector topology is connected. What fails

Metrizable topological vector space - Misplaced Pages Continue

2688-586: Is well defined because for any x ′ ∈ [ x ] {\displaystyle x'\in [x]} we have that d ( x , x ′ ) = 0 {\displaystyle d(x,x')=0} and so d ( x ′ , y ) ≤ d ( x , x ′ ) + d ( x , y ) = d ( x , y ) {\displaystyle d(x',y)\leq d(x,x')+d(x,y)=d(x,y)} and vice versa. Then d ∗ {\displaystyle d^{*}}

2752-1059: The j th {\displaystyle j^{\text{th}}} element of n ∙ {\displaystyle n_{\bullet }} (all other elements of n ∙ {\displaystyle n_{\bullet }} are transferred to m ∙ {\displaystyle m_{\bullet }} unchanged). Observe that ∑ 2 − n ∙ = ∑ 2 − m ∙ {\displaystyle \sum 2^{-n_{\bullet }}=\sum 2^{-m_{\bullet }}} and ∑ U n ∙ ⊆ ∑ U m ∙ {\displaystyle \sum U_{n_{\bullet }}\subseteq \sum U_{m_{\bullet }}} (because U n i + U n j ⊆ U n i − 1 {\displaystyle U_{n_{i}}+U_{n_{j}}\subseteq U_{n_{i}-1}} ) so by appealing to

2816-440: The d {\displaystyle d} -topology on X {\displaystyle X} makes X {\displaystyle X} into a topological group). Conversely, if p {\displaystyle p} is a value on X {\displaystyle X} then the map d ( x , y ) := p ( x − y ) {\displaystyle d(x,y):=p(x-y)}

2880-414: The discrete topology on any non-trivial vector space makes addition and negation continuous but do not make scalar multiplication continuous. If X {\displaystyle X} is an additive group then we say that a pseudometric d {\displaystyle d} on X {\displaystyle X} is translation invariant or just invariant if it satisfies any of

2944-596: The quotient space of X {\displaystyle X} by this equivalence relation and define d ∗ : ( X / ∼ ) × ( X / ∼ ) ⟶ R ≥ 0 d ∗ ( [ x ] , [ y ] ) = d ( x , y ) {\displaystyle {\begin{aligned}d^{*}:(X/\sim )&\times (X/\sim )\longrightarrow \mathbb {R} _{\geq 0}\\d^{*}([x],[y])&=d(x,y)\end{aligned}}} This

3008-924: The article on sublinear functionals , f {\displaystyle f} is uniformly continuous on X {\displaystyle X} if and only if f {\displaystyle f} is continuous at the origin. If all U i {\displaystyle U_{i}} are neighborhoods of the origin then for any real r > 0 , {\displaystyle r>0,} pick an integer M > 1 {\displaystyle M>1} such that 2 − M < r {\displaystyle 2^{-M}<r} so that x ∈ U M {\displaystyle x\in U_{M}} implies f ( x ) ≤ 2 − M < r . {\displaystyle f(x)\leq 2^{-M}<r.} If

3072-403: The following equivalent conditions: If X {\displaystyle X} is a topological group the a value or G-seminorm on X {\displaystyle X} (the G stands for Group) is a real-valued map p : X → R {\displaystyle p:X\rightarrow \mathbb {R} } with the following properties: where we call a G-seminorm

3136-527: The induced topologies. That is, a subset A ⊆ X {\displaystyle A\subseteq X} is open (or closed) in ( X , d ) {\displaystyle (X,d)} if and only if π ( A ) = [ A ] {\displaystyle \pi (A)=[A]} is open (or closed) in ( X ∗ , d ∗ ) {\displaystyle \left(X^{*},d^{*}\right)} and A {\displaystyle A}

3200-568: The inductive hypothesis we conclude that ∑ U n ∙ ⊆ ∑ U m ∙ ⊆ U M , {\displaystyle \sum U_{n_{\bullet }}\subseteq \sum U_{m_{\bullet }}\subseteq U_{M},} as desired. It is clear that f ( 0 ) = 0 {\displaystyle f(0)=0} and that 0 ≤ f ≤ 1 {\displaystyle 0\leq f\leq 1} so to prove that f {\displaystyle f}

3264-516: The inequality f ( s x ) ≤ f ( x ) {\displaystyle f(sx)\leq f(x)} for all unit scalars s {\displaystyle s} such that | s | ≤ 1 {\displaystyle |s|\leq 1} is proved similarly. Because f {\displaystyle f} is a nonnegative subadditive function satisfying f ( 0 ) = 0 , {\displaystyle f(0)=0,} as described in

SECTION 50

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3328-405: The map ( x , y ) ↦ x + y {\displaystyle (x,y)\mapsto x+y} ) is continuous at the origin of X × X {\displaystyle X\times X} if and only if the set of neighborhoods of the origin in ( X , τ ) {\displaystyle (X,\tau )} is additive. This statement remains true if

3392-399: The map p ( x ) := d ( x , 0 ) {\displaystyle p(x):=d(x,0)} is a value on X {\displaystyle X} called the value associated with d {\displaystyle d} , and moreover, d {\displaystyle d} generates a group topology on X {\displaystyle X} (i.e.

3456-1164: The origin for this topology consisting of open sets. Suppose that L {\displaystyle {\mathcal {L}}} is a non-empty collection of F -seminorms on a vector space X {\displaystyle X} and for any finite subset F ⊆ L {\displaystyle {\mathcal {F}}\subseteq {\mathcal {L}}} and any r > 0 , {\displaystyle r>0,} let U F , r := ⋂ p ∈ F { x ∈ X : p ( x ) < r } . {\displaystyle U_{{\mathcal {F}},r}:=\bigcap _{p\in {\mathcal {F}}}\{x\in X:p(x)<;r\}.} The set { U F , r   :   r > 0 , F ⊆ L , F  finite  } {\displaystyle \left\{U_{{\mathcal {F}},r}~:~r>0,{\mathcal {F}}\subseteq {\mathcal {L}},{\mathcal {F}}{\text{ finite }}\right\}} forms

3520-510: The origin in X {\displaystyle X} then d ( x , y ) := f ( x − y ) {\displaystyle d(x,y):=f(x-y)} is a metric defining the vector topology on X . {\displaystyle X.} Assume that n ∙ = ( n 1 , … , n k ) {\displaystyle n_{\bullet }=\left(n_{1},\ldots ,n_{k}\right)} always denotes

3584-447: The positive real numbers, forms a basis for a topology on X {\displaystyle X} that is called the d {\displaystyle d} -topology or the pseudometric topology on X {\displaystyle X} induced by d . {\displaystyle d.} Pseudometrizable space A topological space ( X , τ ) {\displaystyle (X,\tau )}

3648-462: The positive reals, form a neighborhood basis at the origin for this topology consisting of closed set. Similarly, the balanced sets { x ∈ X   :   p ( x ) < r } , {\displaystyle \{x\in X~:~p(x)<;r\},} as r {\displaystyle r} ranges over the positive reals, form a neighborhood basis at

3712-425: The pseudometric d ( x , y ) = p ( x − y ) {\displaystyle d(x,y)=p(x-y)} . This is a convex function of an affine function of x {\displaystyle x} (in particular, a translation ), and therefore convex in x {\displaystyle x} . (Likewise for y {\displaystyle y} .) Conversely,

3776-472: The pseudometric induces an equivalence relation , called the metric identification , that converts the pseudometric space into a full-fledged metric space . This is done by defining x ∼ y {\displaystyle x\sim y} if d ( x , y ) = 0 {\displaystyle d(x,y)=0} . Let X ∗ = X / ∼ {\displaystyle X^{*}=X/{\sim }} be

3840-721: The set of all U i {\displaystyle U_{i}} form basis of balanced neighborhoods of the origin then it may be shown that for any n > 1 , {\displaystyle n>1,} there exists some 0 < r ≤ 2 − n {\displaystyle 0<r\leq 2^{-n}} such that f ( x ) < r {\displaystyle f(x)<r} implies x ∈ U n . {\displaystyle x\in U_{n}.} ◼ {\displaystyle \blacksquare } If X {\displaystyle X}

3904-582: The topology. A topological space is said to be a pseudometrizable space if the space can be given a pseudometric such that the pseudometric topology coincides with the given topology on the space. The difference between pseudometrics and metrics is entirely topological. That is, a pseudometric is a metric if and only if the topology it generates is T 0 (that is, distinct points are topologically distinguishable ). The definitions of Cauchy sequences and metric completion for metric spaces carry over to pseudometric spaces unchanged. The vanishing of

SECTION 60

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3968-626: The translation-invariant trivial metric on X {\displaystyle X} defined by d ( x , x ) = 0 {\displaystyle d(x,x)=0} and d ( x , y ) = 1  for all  x , y ∈ X {\displaystyle d(x,y)=1{\text{ for all }}x,y\in X} such that x ≠ y . {\displaystyle x\neq y.} The topology τ {\displaystyle \tau } that d {\displaystyle d} induces on X {\displaystyle X}

4032-423: The triangle denotes symmetric difference . If f : X 1 → X 2 {\displaystyle f:X_{1}\to X_{2}} is a function and d 2 is a pseudometric on X 2 , then d 1 ( x , y ) := d 2 ( f ( x ) , f ( y ) ) {\displaystyle d_{1}(x,y):=d_{2}(f(x),f(y))} gives

4096-422: The word "neighborhood" is replaced by "open neighborhood." All of the above conditions are consequently a necessary for a topology to form a vector topology. Additive sequences of sets have the particularly nice property that they define non-negative continuous real-valued subadditive functions. These functions can then be used to prove many of the basic properties of topological vector spaces and also show that

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