Tagged Questions

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How do they call the topological tensor product that classifies operators from Hilbert space?

Let $V$ and $W$ be topological vector spaces. There are different ways to complete the tensor product $V \otimes W$, and the only ones that are usually discussed in introductory literature are the ...
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Infinite Dimensional Vector Space: Finite Dim Subspace Closed and Nowhere Dense

Show that any finite-dimensional subspace $(S,\|\cdot\|)$ of an infinite-dimensional normed vector space $(V,\|\cdot\|)$ is closed and nowhere dense. Proof: Let $\{x^{(n)}\}_{n\geq1}$ be a ...
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Prove that the weak$^*$ topology on the space of tempered distributions is not 1st countable

Please, help me with a proof of this (apparently) known fact whose proof is out of my reach, even though I spent a considerable amount of time looking it up: The weak$^*$ topology on the space of ...
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Why do we need dual space [closed]

In functional analysis there are many places where dual space is mentioned, but I still don't understand the real power of that concept. Why do we need the dual space?
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Closed Monoidal Structures On The Category Of Complete Topological Vector Spaces

Context: The category of Banach spaces, with the projective tensor product is a closed monoidal category. Question 1: Is there a tensor product on the category of complete topological vector spaces, ...
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When does a dense subspace destine the weak topology?

Let $E$ be a locally convex space, let $E^{\prime}$ be its continuous dual space and let $F$ be a subspace of $E^{\prime}$ which is dense with respect to the strong topology on $E^{\prime}$ (i.e. the ...
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Weak* continuity

Let $B$ be the open unit ball in $\mathbb{R}^2$ and $\mathcal{M}^+$ the set of nonnegative Radon measures on $B$ and $\mathcal{M}^2$ the set of $\mathbb{R}^2 \text{-valued}$ Radon measures on $B.$ I ...
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Given a Banach space $X$, are weak$^*$ bounded subsets of the dual space $X '$ also strongly bounded (with respect to the usual norm in $X '$)?

Some related facts I already know: 1) In a Banach space $X$, weakly bounded sets are strongly bounded and vice-versa (Thm 3.18 - "Functional Analysis", Rudin); 2) From 1, it follows that my question ...
Let $X$ be a normed vector space, not necessarily Banach. Suppose that $X$ is not reflexive, implying the existence of such $\varphi\in X^{**}$ ($X^{**}$ being the double dual of $X$) of that for any ...