Questions tagged [functional-analysis]

Functional analysis, the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

32,041 questions
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Different topologies in Sobolev space $W^{1,p}$

In paper [1], L.Ambrosio talks about the space $W^{1,p}(\Omega),\ 1\leq p<+\infty$ endowed with four different topologies: The strong topology, denoted by $W^{1,p}(\Omega)$. The weak topology, ...
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Proving one form of Ito Isometry using Functional Analysis

I would like to know whether it is possible to give a proof of (one form of) Ito Isometry using a tool which I like to call "the functional analysis"-way. Let me explain the settings first. What we ...
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The span of a finite number of vectors in a normed vector space is closed

I want to prove that the span $S$ of a finite number of vectors $v$ in an arbitrary normed vector space $V$ is a closed set. My plan is to show that all convergent sequences ${x_n}$ contained in the ...
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Simple Consequences of Goldstine's Theorem

For a normed space $X$, let $J : X \to X^{**}$ be the natural embedding of $X$ into $X^{**}$, and let $B_X$ and $B_{X^{**}}$ denote the closed unit balls of $X$ and $X^{**}$ ...
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Absolutely continuous Banach space valued function

Let $X$ be a Banach space and $F:[a,b] \to X$ be an absolutely continuous function. Is it true that $F$ is differentiable almost everywhere? In particular, for any $f \in L^1([a,b],X)$, is the ...
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Landau inequality for several variables

For $f \in C^n(\mathbb{R})$ and $0 < \alpha < n$, Landau-Kolmogorov inequlity is geven by $$\|f^{(\alpha)}\| \leq K(n,\alpha)\| f\|^{1-\alpha/n}\|f^{(n)}\|^{\alpha/n}, 0 < \alpha < n,$$ ...
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$G$ has Kazhdan's property (T) $\iff$ $G$ has a Kazhdan pair

A locally compact group $G$ is said to be Kazhdan or have Property (T) if for any unitary representation $\rho$ that has almost invariant vectors (a.i.v) it has an invariant vector. Meaning of a.i.v -...
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Krein - Smulian theorem - norm or weak closure?

The theorem of Krein - Smulian reads as follows: The closed convex hull of a weakly compact subset of a Banach space is weakly compact. We consider the closed convex hull - but is it norm closure or ...
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Are $(\ell^\infty(\mathbb{Z}))^*\simeq (\ell^\infty(\mathbb{N}))^*$ isomorphs?

Are $(\ell^\infty(\mathbb{Z}))^*$ and $(\ell^\infty(\mathbb{N}))^*$ isomorphs? I think that I could establish the next function $\Phi:(\ell^\infty(\mathbb{N}))^*\to(\ell^\infty(\mathbb{Z}))^*$ such ...
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Existence of functionals on $L^0$

Studying a paper about risk measures by F. Delbaen, I bumped into this statement: Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space: if $\mathbb{P}$ is atomless, then there exists no ...
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Is the approximate point spectrum simply the union of the essential and point spectra?

I'm refreshing functional analysis from "Hilbert space operators in quantum physics" by Blank, Exner and Havlíček (Springer, 2008). They define essential spectrum for any closed operator on a Hilbert ...
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Can you have an infinite descending chain of dual spaces?

If a Banach space $X$ is not reflexive, then you have an infinite ascending chain of (continuous) dual spaces: $X’$, $X’’$, $X’’’$, etc. None of these are isomorphic to each other or to $X$. My ...