# Questions tagged [topological-vector-spaces]

A topological vector space is a vector space $V$ over a topological field $\mathbb{K}$ (typically $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$), together with a topology on $V$ such that vector addition and scalar multiplication are both continuous. Hilbert spaces and Banach spaces are examples of topological vector spaces.

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### Can we say that $f$ admits a $m(X,X^*)$-continuous extension to $X$?

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### If $X$ is separable space then $X^∗$ is separable in all topologies $\tau$ such that $(X^∗,\tau)^∗ =X$?

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### On Köthe sequence spaces

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### Spaces that are comparable with their compacts

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### A complete metric space with some convex-type property

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### Infra-Pták space that is not Pták

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### On the equivalence of the two definitions of Hadamard differentiability

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### $GL_1(\mathcal{E}'(\mathbb{R}))$ open in $\mathcal{E}'(\mathbb{R})$?

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### Equivalence of σ-convex hull and closed convex hull

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### A topological vector space $X$ is separable if its dual space $X^*$ is separable?

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### Sobolev topology on essentially compactly supported Sobolev-“functions”

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### Metrizability of a topological vector space where every sequence can be made to converge to zero

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### Show that $\big(s(. |C_n)\big)_n$ is equicontinuous on $X^*$

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### Continuous function on colimit

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### Criterion for optimality in two-step optimization procedure

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### Criterion for weak convergence of sequences

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### Pointwise vs. local homotopy equivalences of continuous and smooth complexes of real vector bundles

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### Convergence of sesqui-holomorphic kernels on the diagonal

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### Dual fixed point

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### Inductive limit of $\mathbb R^n$s is Hausdorff and second countable?

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### Do we have $M\hat{\otimes}_A N\cong M\otimes_A N$ if $M$ is a finitely generated projective $A$-module over a nuclear Frechet algebra $A$?

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### Limit of balls in $L^p$

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### Colimits in the category of (not necessarily locally convex) topological vector spaces

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### Duality of Topological Vector Spaces

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### Integration with values in a topological vector space

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### subspace topology and strong topology

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### Open mapping theorem for complete non-metrizable spaces?

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### Every linear topological space embeds into the Tychonoff product of linear metric spaces

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### Reference for “topological affine spaces”

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### Complete dual of bornological space

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### Reference request for Grothendieck's work on “Integration with values in a topological group”

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### A characterization of nuclear functionals in terms of continuity with respect to some special topologies on $B(X)$?

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### Continuous and bornological Hochschild homology

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### When does the dual to the space $K(X)$ of compact operators consist of nuclear functionals?

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### When is a totally bounded set of an inductive limit contained in a component of this limit?

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### Extension of Vector Field in the $\mathcal{C}^r$ topology

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### Is a set of nuclear functionals equicontinuous in compact-open topology if it is equicontinuous on each compact set?

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### Two notions of boundedness in metrizable topological vector space [closed]

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### Is each cometrizable space a subspace of a cometrizable topological group?

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### Spectral theory without topology

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### Invariant compact in division ring

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### Mackey topology characterising property

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### Openness of invertibility in Fréchet spaces for families parameterized by compact spaces

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### Gelfand-Pettis integral: what does it mean for a topological vector space to “admit a dual space?”

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### Is the compact-open topology on the dual of a separable Frechet space sequential?

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### The completeness of spaces of continuous functions with the compact-open topology

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### On convergent sequences in locally convex topological vector spaces

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### References on categorical TVS theory

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### Is restriction a closed map?

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