# 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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### Critical point for composite function implies $f'(x_{0}) = 0$ in locally convex spaces

Let $X, Y$ be locally convex spaces over $\mathbb{R}$ and $f: X \to Y$. Suppose $f$ is Fréchet differentiable at a point $x_{0} \in X$. Given an interval $0 \in I \subset \mathbb{R}$, we call a smooth ...
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### Textbook on TVS that contains Theorem 32.2 about the completeness of the space of continuous linear maps

I have come across this thread about TVS. The OP took below screenshot of THEOREM 32.2. Could you please elaborate on the book from which the screenshot was taken?
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### Existence of measure with given Fourier transform

Bogachev's "Gaussian measures" contains the following theorem: Theorem 3.3.1: Let $\mathcal{X}$ be a locally convex space and $G\subset \mathcal{X}^\ast$ a linear subspace separating the ...
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Let $\left(\mathcal{X}, \tau\right)$ be a locally convex topological space and $K$ a compact subset. Is there a reference for the proof that on $K$ the relative topology coincides with the relative ...