# 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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### The dual of a dual space with the topology of uniform convergence on compact subsets?

$W$ is a Banach space. The topology of $W^*$ is the uniform convergence on the compact subsets of $W$. That is generated by the family of seminorms $$p_K(f)=\sup_{x\in K}|f(x)|,$$ for all compact ...
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### Infinite Dimensional Vector Space: Finite Dim Subspace Closed and Nowhere Dense

Show that any finite-dimensional subspace $(S,\|\cdot\|)$ of an infinite-dimensional normed vector space $(V,\|\cdot\|)$ is closed and nowhere dense. Proof: Let $\{x^{(n)}\}_{n\geq1}$ be a ...
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### On convex hulls and intersections of chains of compact sets

Let $V$ be a topological vector space, let $\{ C_i \}_{i \in I}$ be a set of compact subsets of $V$ which forms a chain with respect to inclusion. For now, assume the following stronger properties: ...
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### Prove that the weak$^*$ topology on the space of tempered distributions is not 1st countable

Please, help me with a proof of this (apparently) known fact whose proof is out of my reach, even though I spent a considerable amount of time looking it up: The weak$^*$ topology on the space of ...
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### Linear independence in 3-space

The orthogonal orthonormal basis vectors $i,j,k$ in $3$-space are generally accepted as linearly independent. However $j$ can be derived from $i$, by the process of differentiation, and likewise ...
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### Why do we need dual space [closed]

In functional analysis there are many places where dual space is mentioned, but I still don't understand the real power of that concept. Why do we need the dual space?
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### $\sigma$-Algebra: Why do we want it to contain complements as well?

Everybody Hello, I was always wondering: (Please answers apart from historical reasons) Why do we want a $\sigma$-Algebra to possess more than just its crucial disjoint $\sigma$-union property? Say, ...
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### Closed Monoidal Structures On The Category Of Complete Topological Vector Spaces

Context: The category of Banach spaces, with the projective tensor product is a closed monoidal category. Question 1: Is there a tensor product on the category of complete topological vector spaces, ...
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### When does a dense subspace destine the weak topology?

Let $E$ be a locally convex space, let $E^{\prime}$ be its continuous dual space and let $F$ be a subspace of $E^{\prime}$ which is dense with respect to the strong topology on $E^{\prime}$ (i.e. the ...
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### Weak* continuity

Let $B$ be the open unit ball in $\mathbb{R}^2$ and $\mathcal{M}^+$ the set of nonnegative Radon measures on $B$ and $\mathcal{M}^2$ the set of $\mathbb{R}^2 \text{-valued}$ Radon measures on $B.$ I ...
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### Given a Banach space $X$, are weak$^*$ bounded subsets of the dual space $X '$ also strongly bounded (with respect to the usual norm in $X '$)?

Some related facts I already know: 1) In a Banach space $X$, weakly bounded sets are strongly bounded and vice-versa (Thm 3.18 - "Functional Analysis", Rudin); 2) From 1, it follows that my question ...
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### Two basic questions about topological linear space theory

For a topological vector space(tvs), I'd like to know whether 1.there exist a topological vector space V which is a Hausdorff space but does not satisfies the first countable axiom or 2.there exist ...
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### Is Reflexivity Necessary for the Weak and Weak* Topologies to Coincide?

Let $X$ be a normed vector space, not necessarily Banach. Suppose that $X$ is not reflexive, implying the existence of such $\varphi\in X^{**}$ ($X^{**}$ being the double dual of $X$) of that for any ...
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### Is there a topological space and meanwhile a linear space such that its vector addition is discontinuous but scalar multiplication is continuous?

The title is the question. Does there exist a topological space and meanwhile a linear space X such that its vector addition operation is discontinuous but scalar multiplication operation is ...
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### Example of a sequence of functions

Construct an example of a sequence of functions $(f_n)$ defined on $[0,1]$ such that $f_n$ converges pointwise to $0$ and for every sequence of numbers $(a_n)$ that tends to $\infty$, sequence ...
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### Question on the proof that $C(\Omega)$ is a Frechet space

I am using Rudin's book on Functional Analysis. I am studying the proof that the space $C(\Omega)$ of continuous functions on an open set $\Omega \subseteq \mathbb{C}$ is a Frechet space. I ...
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### Constructing a closed, convex subset of $X^{\ast}$ that is not weakly-* closed

I'm asked to show that if $X$ is a non-reflexive Banach space, there exists (norm) closed and convex subsets of $X^\ast$ that are not $w^{\ast}$-closed. In other words, there's no analogue of Mazur's ...
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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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### How general is the convergence of the exponential function's power series?

Let $\mathbf{V}$ be a Fréchet space whose underlying set is $V$. Let $\;\; \beta \: : \: V\times V \: \to \: V \;\;$ be a continuous bilinear map that has an identity element and is ...
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### How can we ensure that a space is a subset of locally convex topological space?

I am looking for fast ways to ensure that a given set is a subset of topologically locally convex space. I have already read the posts post1:seminorms-in-locally-convex-spaces, ...
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### Notions of topological spaces and matrices

How can I show that: For all $r>0$, exists $A$ nonsingular matrix, such that $B_r(A)\subseteq GL_n(\mathbb{C})$ For all $A\in GL_n(\mathbb{C})$ and for all $r>0$, there exists $\alpha>0$ ...
A continuum is a compact connected metric space. A plane continuum is a continuum contained in $R^2$. We say that a plane continuum does not separate the plane provided that $R^2\setminus X$ is ...
### if $v$ is a member of $H$ and $v$ is not a member of $M$ then $u$ is member of $K$. How is this possible?
Let $(V,K)$ and $u,v$ is a member of $V$. Suppose that $M$ is a subset of $V$ is a subspace of $V$ with basis $B_m=\{m_1,...,m_r\}$ with $r$ less than and equal to $n$. Let $H$ be a subspace spanned ...