# 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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### Are the finite signed measures on a compact set $M(Compact)$ first countable?

Let $M(Compact)$ be the set of finite signed measures on a countable set? (with the topology generated by the sets $\left\{ \mu \in M(Compact) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$...
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### How do they call the topological tensor product that classifies operators from Hilbert space?

Let $V$ and $W$ be topological vector spaces. There are different ways to complete the tensor product $V \otimes W$, and the only ones that are usually discussed in introductory literature are the ...
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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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### Is there a non-complete and non-separate metric space?

Is there a (non-trival) non-complete and non-separate metric space? Some notions are here: math.stackexchange.com/questions/182316.
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### Proof details for the first theorem in functional analysis

The Proof details for the first theorem in functional analysis is given in this website: I have one questions for the proof of lemma 5 given here. (1) Why this claim hold? from equality $0+0=0$ ...
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### In a topological vector space, show if $A$ and $B$ are bounded, then $A + B$ is bounded?

I get as far as this before I am stuck: Pick any neighbourhood of $0$ and call it $U$. Then there exists $a, b$ such that $A \subseteq aU$ and $B \subseteq bU$. So hence $A + B \subseteq aU+bU$. ...
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### How to prove in a topological vector space: cl(A) + cl(B) is a subset of cl(A+B), where cl denotes closure?

I'm not sure where to really proceed. My process is as follows. Take any $x \in cl(A)+cl(B)$. Assume for a contradiction that $x \notin cl(A+B)$. Then there exists an open set $U$ such that $x \in U$,...
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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 power-...
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 $\alpha>0$ ...