Questions tagged [locally-convex-spaces]

For questions about topological vector spaces whose topology is locally convex, that is, there is a basis of neighborhoods of the origin which consists of convex open sets. This tag has to be used with (topological-vector-spaces) and often with (functional-analysis).

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Why the space of section of a vector bundle is complete?

This page is about the space of sections: Let $E \stackrel{\mathrm{fb}}{\rightarrow} \Sigma$ be a smooth vector bundle. On its real vector space $\Gamma_{\Sigma}(E)$ of smooth sections consider the ...
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Conditions for locally convex space to be normable

Let $X$ be a Hausdorff locally convex space and $P$ is a family of seminorms on $X$. How to show that $X$ is normable iff $P$ is equivalent to a finite subfamily $P_0 \subset P$? One implication is ...
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Discontinuous linear operator.

Does there exists a discontinuous linear operator $T$ between Hausdorff locally convex spaces $X$ and $Y$ such that $T$ is continuous with respect to the weak topologies on $X$ and $Y$?
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Topology of compact convergence on space of holomorphic functions

Consider $\mathbb{D}_{R} = \{z \in \mathbb{C}: |z| < R\}$. For given $f \in \mathscr{O}(\mathbb{D}_{R})$ - space of holomorphic functions on the open disk, denote by $c_n$ the $n$-th Taylor's ...
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Smallest vector topology

This is with regard to this question: Topology induced by seminorms and initial topology I saw somwehere that topology $\mathcal{S}$ is the smallest topology with respect to which all the seminorms ...
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Equivalence of two seminorms

Let us consider the space $C^0([0,1])$ of real continuous functions over $[0,1]$. Let $A = \{a_n : n \in \mathbb{N}\}$ be a countable set of $[0,1]$ and $\alpha_n$ be strictly positive real numbers ...
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Linear functional on LCS is continuous iff it is bounded by finite linear combination of topology-defining semi-norms

Consider a linear functional f on a TVS, whose topology is generated by a family of semi-norms $\mathcal{P}$, such that the topology is Hausdorff. In functional analysis by Conway, it is said that f ...
I'm currently working through Conway's Functional Analysis, and have stumbled across a result that I can't seem to find the justification for myself: Let $\mathcal{X}$ be a Real Locally Convex Space ...