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Let $X$ and $Y$ be topological linear spaces which are complete & Hausdorff, and admit dual spaces which separate points. Suppose the topologies are non-separable and non-metrizable. Let $f : X \to Y …

Let $U_\infty$ be a compact space, and let $U_r$ be an increasing family of compact subspaces whose closure is all of $U_\infty$. That is, $U_r \subseteq U_{r'}$ if $r \le r'$ and $U_\infty = \overli …

No. Bounded means continuous, for linear operators. If a linear operator is bounded (continuous) on a dense subset, then you can extend it continuously to the whole space, which means it is bounded. …

Is there an example of a locally convex topological vector space which is not compactly generated? (any such example must be non-Fréchet, since all Fréchet spaces are compactly generated) (note: I a …

From Wikipedia: In mathematics, in particular in mathematical analysis, the Whitney extension theorem is a partial converse to Taylor's theorem. Roughly speaking, the theorem asserts that if $A$ i …

Let $T \subseteq \mathbb R$ be a closed set of real numbers. Let $X := C(T, \mathbb R)$ denote the Fréchet space of continuous real-valued functions on $T$. The topology on $X$ is generated by seminor …

Let $X$ denote a topological affine space (with no additional assumptions). Let $X^*$ denote its dual space of continuous affine functionals, equipped with the weak-$*$ topology. It is easy to see tha …

Thanks William for reaching out (and thanks David Roberts for the hat tip to my talk). Let me give an intentionally fuzzy, high-level answer. Generally speaking, the Yoneda Lemma allows you to make a …

Let $f : X \to Y$ be a continuous map of complete topological vector spaces. Suppose that $A \subseteq X$ and $B \subseteq Y$ are proper, dense linear subspaces, and that the restriction map $f : A \t …

Let $X$ be a separable Banach space, and let $\theta \mapsto \mathbb P_\theta$ be a weakly-continuous family of Radon probability measures. Suppose that each $\mathbb P_\theta$ admits a mean vector $\ …

Let $H$ be a Hilbert space, and let $X$ be a topological vector space. Under what conditions on the topologies of $X$ and $H$ does there exist an injective, continuous linear map $f : H \to X$? Su …

Here's an answer in the case that $X$ and $Y$ are Gaussians. There's such a rigid algebraic structure there that I don't know whether I'm exploiting that, or a more general property of the distributi …

Let $X$ be a topological affine space over a complete base field $\mathbb S := \mathbb C$, $\mathbb R$ or $\mathbb Q_p$. Let $X^*$ be the dual space of continuous affine functionals equipped with the …

Since a troll bumped this question to the front page, I might as well answer it. The technology which provides the solution is called regular conditional probability or disintegration.

Sorry for the necromancy. Here's an attempt at constructing a $\sigma$-algebra using the tensor product of $\sigma$-algebras. This should likely not result in a Borel structure (i.e., a $\sigma$-algeb …