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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Motivation for the notion of locally convex topological vector space

Is the only motivation for the notion of locally convex topological vector space that the local bases have some nice property i.e. convex, balanced, absorbing ?
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Topological Tensor Product is a Topological Ring Independent of the Choice of Basis

Let $A, B$ be commutative rings containing a field $k$, with $B$ a finite dimensional $k$-module, $w_1, ... , w_N$ a basis. If $w_iw_j = \sum\limits_{n=1}^N c_{ijn}w_n$, then we can define $C$ to be ...
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Balanced Core: Explicit Expression?

Denote the collection of all balanced subsets by: $\mathcal{B}:=\{B\subseteq X: B\text{ balanced}\}$ Since the union of arbitrary balanced sets is balanced we can form the balanced core of arbitrary ...
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Properties of compact set: non-empty intersection of any system of closed subsets with finite intersection property

Let $X$ be a Hausdorff topological vector space. Let $C$ be a nonempty compact subset of $X$ and $\{C_\alpha\}_{\alpha \in I}$ be a collection of closed subsets such that $C_\alpha \subset C$ for each ...
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Counterexample about non Hausdorff topological vector spaces

I have some troubles with Hausdorffness in TVS: Question 1. Is there any topological vector space $X$ which is not Hausdorff? Question 2. Give an explicit example of a topological vector space $X$ (...
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Question about proof of Krein-Milman Theorem.

I am in the middle of working through the details of the proof of the Krein-Milman theorem in Rudin's Functional Analysis (Theorem 3.23), and I am stuck on one detail. I will state the theorem and ...
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A topology on a set $X$ is a family $\mathcal{T}$ of subsets of $X$, which are open sets and satisfy: (1) $\emptyset, X \in \mathcal{T}$. (2) Any union of elements of $\mathcal{T}$ belongs to $\... 1answer 183 views Is every “nice” topological vector space a manifold? Say$V$is a topological vector space. What conditions do you need to add on$V$to make it a (topological, maybe infinite-dimensional) manifold? For instance, can we view the Schwartz class ... 1answer 85 views Every Bounded set contained in a Compact set In a general metric space, is every bounded set contained in a compact set? 0answers 66 views topological vector space of measure functions Let$(X, \mathcal X, \mu )$be a measure space, and let$ L(X)$be the space of measurable functions$f: X \to \mathbb C$. Show that the sets$B(f, \epsilon ,r ): = \{ g \in L(X) : \mu( \{ x : | f(x) -...
Let $V$ be a vector space, and let $(\mathcal F_ \alpha ) _{ \alpha \in A}$ be a family of topologies on V, each of which turning $V$ into a topological vector space. Let $\mathcal F$ be the vector ...
Let $E$ be a Banach space, $A_n$ be an increasing sequence of finite dimensional subspaces of $E$, $B_n$ be an increasing sequence of subspaces of $A_n$ and let $C_n = A_n/B_n$. Assume that the ...