Questions tagged [topological-vector-spaces]
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.
1,897 questions
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Why locally convex space is usually required to be hausdorff
Recently I'm reading Conway's 'A Course in Functional Analysis', in his book, the definition of LCS (locally convex space) is as followed
It seems that for basic definition of LCS, we do not need the ...
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Completeness of inductive limits
Let $X$ be the inductive limit of an increasing sequence $(X_n)$ of locally convex subspaces, as defined here. If each $X_n$ is Hausdorff and complete, then so is $X$. The linked post contains a proof ...
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Is Topology on Uniform Convergence on Compacts Unique
Looking at Example 1.44 in Rudin's functional analysis https://59clc.wordpress.com/wp-content/uploads/2012/08/functional-analysis-_-rudin-2th.pdf , it can be shown that this Frechet space topology ...
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Does inductive limit topology with cofinal chain satisfy the subspace property?
Let $\{ (E_\alpha, \tau_\alpha) \}_{\alpha\in A}$ be a family of locally convex spaces, where each $E_\alpha$ is a vector subspace of a vector space $E$ over $\mathbb K.$ Suppose further that the $E_\...
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Possible equivalent definition of the topology on the set of Schwartz functions and Schwartz distributions
I have the following definition of the Schwartz space $\mathcal{S}(\mathbb{R}^d)$: for each $\alpha, \beta \in \mathbb{N}^d$ one defines the semi-norm $|\varphi|_{\alpha, \beta} = \text{sup} |y^{\beta}...
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Given a judiciously chosen function space, is it possible to construct a Hilbert space as a $L^2$ space over this space?
I'm told that given a measure space $ (X,d\mu)$ the space of square integrable functions (modulo sets of measure zero), $L^2(X,d\mu)$ is always a Hilbert space.
My question is this: Is it possible to ...
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Vector topologies and lattice operations
Let $X$ be a vector space and $\mathcal T(X)$ the set of all possible (not necessarily linear) topologies on $X$ (a complete lattice w.r.t. the 'sup' and 'inf' operations, where the 'inf' is explictly ...
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Proof verification : every strict LF space is complete
It is well-known that every strict LF space $X$ is complete, i.e., every Cauchy filter on $X$ is convergent. However, the only proofs that I could find online are faithful copies of Trèves' proof in ...
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Local base for inductive limit topology
The following is based on Schaefer & Wolff's Topological Vector Spaces, 2nd edition, §II.6. Let $E$ be a vector space, $\{E_\alpha\}_{\alpha\in A}$ be locally convex spaces (not necessarily ...
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Why LF-continuity in the definition of distributions?
A distribution is a continuous linear functional on the space $\mathcal D$ of smooth functions $f : \mathbb R \to \mathbb R$ with compact support. The continuity is with respect to a particular ...
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Measurability of convex hulls in locally convex spaces
(Posted also on MathOverflow.)
Let $X$ be a locally convex space over $\mathbb{R}$, and let $X^*$ denote its topological dual. The cylindrical $\sigma$-algebra $\mathcal E(X)$ is defined as the ...
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Linearly compact modules with topologies weaker than the discrete topology
I'm trying to ramp up on linearly compact modules (linearly compact rings, actually) and I wonder in what situations we can expect to find or use something other than the discrete topology. It seems ...
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If $(x_n)$ is weakly unconditionally Cauchy in a Hausdorff locally convex space $X$ and $(a_n)\in c_0$ then $\sum_n a_n x_n$ is Cauchy in $X$
I read in a research paper On Schwartz's C-spaces and Orlicz's O-spaces by S. Díaz Madrigal that if the sequence $(a_n)$ converges to zero and $(x_n)$ is a weakly unconditionally Cauchy sequence in a ...
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What topology on distributions makes the Dirac embedding continuous?
I have a fairly limited knowledge of functional analysis, being more of algebraist-geometer by training.
If some not-too-technical textbooks, suitable for a non-analyst like me, come to your mind, ...
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Is every quotient of a Smith space by a closed subspace again complete?
A Smith space $X$ is a complete locally convex Hausdorff topological vector space such that there exists an absolutely convex compact subset $K \subseteq X$ with $X = \bigcup_{c > 0} c K$ and such ...
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