# Tagged Questions

The study of vector space with a topology which makes the maps which sums two vectors and which multiply a vector by a scalar continuous. It's a natural generalization of normed spaces.

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### “Every linear mapping on a finite dimensional space is continuous”

From Wiki Every linear function on a finite-dimensional space is continuous. I was wondering what the domain and codomain of such linear function are? Are they any two topological vector ...
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### The kernel of a continuous linear operator is a closed subspace?

If $V$ and $W$ are topological vector spaces (and $W$ is finite-dimensional) then a linear operator $L\colon V\to W$ is continuous if and only if the kernel of $L$ is a closed subspace of $V$. Why ...
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### Weak *-topology of $X^*$ is metrizable if and only if …

Let $X$ be a topological vector space on which $X^*$ separates points. Prove that "the weak *-topology of $X^*$ is metrizable if and only if $X$ has a finite or countable Hamel basis"? (A set $\beta$ ...
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### Dual of a dual cone

Any hint on how to prove the following please: Let $K$ be a convex cone, and $K^*$ its dual cone. Prove that $K^{**}$ is the closure of $K$. Thanks!
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### Do continuous linear functions between Banach spaces extend?

Just wondering... Let $E$, $G$ be Banach spaces, let $U\subset E$ be a subset of $E$, and let $f:U\rightarrow G$ be a continuous linear function. Can $f$ be extended to a continuous linear function on ...
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### How to endow topology on a finite dimensional topological vector space?

This post may be coincide with some of the contents here. From Conway, A course in functional analysis, page 104. If $H$ is a finite dimensional vector space and $F_{1},F_{2}$ are two topologies ...
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### Is any Banach space a dual space?

Let $X$ be a Banach space. Is there always a normed vector space $Y$ such that $X$ and $Y^*$ are isometric or isomorphic as topological vector spaces (that is, there exists a linear homeomorphism ...
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### If weak topology and weak* topology on $X^*$ agree, must $X$ be reflexive?

Let $X$ be a Banach space and suppose that the weak topology on $X^*$ agrees with the weak* topology on $X^*$. Must $X$ be reflexive? To prove the contrapositive, it will suffice to assume that $X$ ...
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### Part (a) of Exercise 13 of first chapter of Rudin's book “Functional Analysis”

I would really appreciate it if you could give me some advice on the part (a) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis": Let $C$ be the vector space of all ...
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### $\omega$ - space of all sequences with Fréchet metric

I'm working on to prove the following: Show that the convergence in the space $\omega$ (space of all sequences with respect to the Fréchet metric) is the coordinate convergence. Any hint is ...
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### When is a notion of convergence induced by a topology?

I'm interested in sufficient conditions for a notion of sequential convergence to be induced by a topology. More precisely: Let $V$ be a vector space over $\mathbb{C}$ endowed with a notion $\tau$ of ...
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### What is the topological dual of a dual space with the weak* topology?

I'm trying to understand a claim I heard in class. To be concrete, suppose $X$ is a compact, hausdorff topological space, and let $C(X)$ be the space of continuous functions on $X$ with the supremum ...
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### Sum of closed and compact set in a TVS

I am trying to prove: $A$ compact, $B$ closed $\Rightarrow A+B = \{a+b | a\in A, b\in B\}$ closed (exercise in Rudin's Functional Analysis), where $A$ and $B$ are subsets of a topological vector space ...
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### Part (b) of Exercise 13 of first chapter of Rudin's book “Functional Analysis”

I would really appreciate it if you could give me some advice on the part (b) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis": Let $C$ be the vector space of all ...
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### How can I prove this is a metric?

While proving the Banach-Alaoglu theorem, one needs to prove the topology induced by a countable family of seminorms $\rho_n$ on a vector space $X$ is metrizable if $X$ is Hausdorff with that topology....
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### When do weak and original topology coincide?

Let $X$ be a topological vector space with topology $T$. When is the weak topology on $X$ the same as $T$? Of course we always have $T_{weak} \subset T$ by definition but when is $T \subset T_{weak}$?...
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### Does every $\mathbb{R},\mathbb{C}$ vector space have a norm?

Is there a canonical way to define on any vector space over $\mathbb{K}=\mathbb{R},\mathbb{C}$ a norm ? (Or, if there isn't, can someone give me an example of a vector space over $\mathbb{K}$ that is ...
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### A surjective linear map into a finite dimensional space is open

I'm in search of different proofs of the following proposition: $\bf{Proposition}$: Suppose $X$ and $Y$ be topological vector spaces, $\text{dim }Y<\infty$, and $\Lambda:X\to Y$ is a surjective ...
For a (possibly infinite-dimensional) vector space $V$, I thought about the following topology $\tau$: Let $O \in \tau$ if every $x \in O$ has the property that for every $v \in V$, there is an $\... 1answer 551 views ### An explicit construction for a “doubly weak” topology Let$(X,s)$be a topological vector space over$\mathbb{F}$with linear topology$s$, which we will henceforth refer to as the strong topology. Then, as usual we can construct the continuous dual ... 2answers 709 views ### Meaning of “a mapping preserves structures/properties” Sometimes I see something like "a mapping preserves the structures of its domain and of its codomain". From Wiki about morphisms in category theory: a morphism is an abstraction derived from ... 3answers 312 views ### Can continuity of inverse be omitted from the definition of topological group? According to Wikipedia, a topological group$G$is a group and a topological space such that $$(x,y) \mapsto xy$$ and $$x \mapsto x^{-1}$$ are continuous. The second requirement follows from the ... 3answers 751 views ### Topology on the general linear group of a topological vector space Let$K$be a topological field. Let$V$be a topological vector space over$K$(if it makes things convenient, you may assume it is finite dimensional). Naive Question: Is there a canonical way of ... 0answers 143 views ### What do we call a Schauder-like basis that is uncountable? In a topological vector space, every Schauder basis is assumed countable, by definition. Supposing we drop the countability condition, we call this a [what goes here?] basis? 1answer 52 views ### How to construct a particular convex set (when defining inductive limits of Frechet spaces in Reed and Simon) Let$X$be a topological vector space (over$\mathbb{C}$) whose topology is defined by a family of separating seminorms$\{\rho_\alpha\}$. Let$X_1$be a vector subspace of$X$whose topology is the ... 1answer 95 views ### Part (c) of Exercise 13 of first chapter of Rudin's book “Functional Analysis” I would really appreciate it if you could give me some advice on the part (c) of Exercise 13 of first chapter of Walter Rudin's book "Functional Analysis": Let$C$be the vector space of all ... 1answer 83 views ### TVS: Uniform Structure Disclaimer: This thread is meant informative and therefore written in Q&A style. Of course everybody is encouraged to give an answer as well! Prove that any topological vector space gives rise ... 1answer 86 views ### About a weak topology on TVS Let$(X,\tau)$be a topological vector space and suppose$P$is a separating family of seminorms on$X$. Denote by$\sigma(X,P)$, the weak topology on$X$, the smallest topology on$X$that makes each ... 1answer 59 views ### Lebesgue measure as$\sup$of measures of contained compact sets I know, from Kolmogorov-Fomin's Элементы теории функций и функционального анализа, the definition of external measure of a bounded set$A\subset \mathbb{R}^n$as $$\mu^{\ast}(A):=\inf_{A\subset \... 0answers 52 views ### Doubt Concerning the Definition of a Locally Convex Space Structure through Seminorms? In the book Introduction to Functional Analysis written by A. E. Taylor there are the following theorems: Theorem 1. Suppose that X is a linear space and that \mathscr{U} is a nonempty family ... 1answer 2k views ### Semi-Norms and the Definition of the Weak Topology When I was searching for the definition of the weak topology, I found two different definitions. One defines the weak topology in terms of a family of semi-norms, while the other defines it in terms ... 1answer 961 views ### Hahn-Banach theorem: 2 versions I have a question regarding the Hahn-Banach Theorem. Let the analytical version be defined as: Let E be a vector space, p: E \rightarrow \mathbb{R} be a sublinear function and F be a subspace of ... 1answer 240 views ### Why do we give C_c^\infty(\mathbb{R}^d) the topology induced by all good seminorms? Briefly, my question boils down to the following: What benefits do we gain from considering the space of test functions in the topology induced by all "good" seminorms, as opposed to other topologies ... 1answer 681 views ### Sequential and topological duals of test function spaces Given a test function space, in particular \mathcal{S}=\mathcal{S}(\mathbb{R}^n) (the Schwartz space) or \mathcal{D}=\mathcal{D}(\mathbb{R}^n) (the space of compactly supported smooth test ... 1answer 240 views ### Is there finest topology which makes given vector space into a topological vector space? I we are given a vector space (V,+,\cdot) over a field \mathbb K (where \mathbb K=\mathbb R or \mathbb K=\mathbb C), is there the finest topology \mathcal T, such that (V,\mathcal T) is a ... 1answer 110 views ### local convexity of L_p spaces wiki says The spaces L_p([0, 1]) for 0 < p < 1 are equipped with the F-norm they are not locally convex, since the only convex neighborhood of zero is the whole space Why is this so? http://... 1answer 165 views ### Relative countable weak^{\ast} compactness and sequences I am finding serious difficulties in understanding some things about relative countable compactness and the use of sequences in proving it by my functional analysis text, Kolmogorov-Fomin's. For ... 1answer 782 views ### proving that SO(n) is path connected Our professor gave us exercise to show that G=SO(n,\mathbb R) is path connected. He gave some hints, using them I have come upto this far: I have shown that SO(n) acts on S^{n-1} transitively ... 1answer 611 views ### Finding the topological complement of a finite dimensional subspace I know that for any finite dimensional subspace F of a banach space X, there is always a closed subspace W such that X=W\oplus F, that is, any finite dimensional subspace of a banach space is ... 2answers 304 views ### Left topological zero-divisors in Banach algebras. Let A be a unital Banach algebra. Define \zeta: A \longrightarrow [0,\infty) by$$ \forall a \in A: \quad \zeta(a) \stackrel{\text{def}}{=} \inf_{b \in \mathbb{S}(A)} \| ab \|, $$where \... 1answer 162 views ### On separable Hilbert space H, weak operator topology is metrizable on bounded parts of B(H) The following is a theorem of Takesaki's operator theory: In this proof, weak topology means weak operator topology. I'm wonder why the theorem holds just for bounded parts of B(H) and also ... 1answer 2k views ### Interior of a convex set is convex [duplicate] A set S in \mathbb{R}^n is convex if for every pair of points x,y in S and every real \theta where 0 < \theta < 1, we have \theta x + (1- \theta) y \in S. I'm trying to show that ... 1answer 394 views ### Uniqueness of the derivative in locally convex topological vector space I need a hint of proof of uniqueness of the derivative in locally convex topological vector space (it's asserted in Lang's "Introduction to differentiable manifolds"). Define derivative of a function ... 1answer 2k views ### Question about proof that finite-dimensional subspaces of normed vector spaces are direct summands I am reading a proof that finite-dimensional subspaces of normed vector spaces have closed direct sum complements. This is the proof: Let \{e_1, ..., e_n\} be a basis for \mathcal M. Every x ... 1answer 352 views ### About Lusin's condition (N) We say that f:[0,1]\to \mathbb{R} satisfies Lusin's condition (N) provided$$m(f(B))=0 \quad\mbox{whenever}\quad B\subseteq [0,1] \mbox{ with }m(B)=0$$where$m$stands for the Lebesgue measure on$\...
Let $(X, \left \| \cdot \right \|_X )$, $(X, \left \| \cdot \right \|_Y)$ two normed vector spaces with $X \subset Y$, by definition we have $X \hookrightarrow Y$ if \$\left \| x \right \|_Y \leq C \...