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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).

Functional analysis is 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, including spectral theory, as well as measure, integration, probability on infinite dimensions, and also manifolds with local structure modeled by these vector spaces.

For basic questions about functions use more suitable tags like , or .

52582 questions
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Weak convergence in $L^p$ and uniform convergence

I don't understand the last line of a proof (which is supposed to be obvious...), could you help me? The context is the following. We have a bounded open set $U$ of $\mathbb{R}^m$, and a $C^\infty$-mapping $$F : \mathbb{R}^m \times \mathbb{R} \times…
Nicolas
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$\|S\|=\sup\{|\langle Sx\;,\;y\rangle |;\;\|x\| \leq 1,\, \|y\| \leq 1\}\,?$

Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $F$. Let $S\in \mathcal{B}(F)$. The norm of $S$ is defined us $$\|S\|:=\sup_{\substack{x\in F\\…
Schüler
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Basic Open Problems in Functional Analysis

Hello I was wondering if there exists open problems in functional analysis that don't require too much machinary for studying them, I mean, problems that don't require high level prerequisites.. Does anyone know any of them:
PtF
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Let $T: E \rightarrow E^*$ be a linear operator satisfying $\langle Tx,x \rangle \geq 0 \forall x \in E$. Prove T is bounded.

I'm trying to use the closed graph theorem, i.e, I'm trying to prove that $Graph(T) = \{ (x,Tx) ; ~x \in E \}$ is closed in $E \times $F, but I'm a little bit confused. So, i'd like some help in proving it. Here, given $f$ function, $\langle f,x…
Bohrer
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Is $\left(\sum\limits_{k=1}^\infty \frac{x_k}{j+k}\right)_{j\geq 1}\in\ell_2$ true if $(x_k)_{k\geq 1}\in\ell_2$

Let $(x_k)_{k\geq 1}\in\ell_2$. Consider $\left(\sum\limits_{k=1}^\infty \dfrac{x_k}{j+k}\right)_{j\geq 1}$. Now my question is that whether $\left(\sum\limits_{k=1}^{\infty}\frac{x_k}{j+k}\right)_{j\geq 1}$ belongs to $\ell_2$ or not. The following…
molan
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Composition of function with linear functional is Lipschitz implies function itself is Lipschitz

This is taken from Conway's A course in functional Analysis (p. 98, Exercise 9): If $(S,d)$ is a metric space and $X$ is a normed space, show that if $f:S\rightarrow X$ is a function such that for all $L\in X^*$ (the continuous dual of $X$) $L\circ…
user3157436
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Topological Vector Space

Show: Let $\{U_{\lambda}: \lambda \in L\}$ be a base of neighbourhoods at $0$ in a topological vector space $\mathcal{X}$. Then $\{U_{\lambda}+ U_{\lambda}: \lambda \in L\}$ is also a base of neighbourhoods. I have an intuition that this is to…
user24367
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The Space $C(\Omega,\mathbb{R})$ has a Predual?

Let $\Omega$ be a compact metric space and $C(\Omega,\mathbb{R})$ the space of borel continuous real valued functions. I would like to know if there is any real Banach space $V$ such that its dual space (topological)is exactly…
Leandro
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Taylor expansion of functional

The question asks for the Taylor expansion of a functional. Thus, given a real functional $f(g(x))$, what is the Taylor expansion about a function $h(x)$. What if the function is multi-variate, e.g., $f(x_1,x_2,g(x_1,x_2))$? I've searched the web…
Jorge
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Rudin 13.3 zero operator as adjoint

For an assignment I have to show that exists a densely defined operator on a infinite dimensional separable Hilbert space, such that its adjoint is the zero operator on the zero subspace. To show this there is a reference to exercise 13.3 in the…
simon
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Examples of (infinite dimensional) linear operators

I'm trying to familiarize myself with linear operators. In finite dimensions it is clear to me that they are matrices. No problem there. But then in infinite dimensions matters are not so clear to me. Of course the identity map is a linear operator.…
user66372
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$X\subsetneqq Y$ but $X^\star=Y^\star$

Are there $X,Y$ real Banach spaces, such that $X\subsetneqq Y$ (strictly contained) and $X^\star=Y^\star$, where $\star$ denotes the topological dual? This property is not true for Hilbert spaces, nor even for $L^p$ spaces, so I was thinking to try…
Tomás
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Hahn Banach Theorem

It is stated often that the Hahn Banach Theorem makes the study of the dual space "interesting". What does this exactly mean though? I.e what is exactly meant by "interesting"? I am puzzled as to why it follows immediately from Hahn-Banach that the…
user58514
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Questions about Bochner integral

I was wondering If there is distinction between existence of Bochner integral and Bochner integrability, or the two always mean the same? If in Bochner integral, the integrand is assumed to be measurable wrt the Borel $\sigma$-algebra of the…
Tim
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Can $\{1, x^2, x^3, x^4, ...\}$ approximate $x$ on $[0,1]$?

Can $\{1, x^2, x^3, x^4, ...\}$ approximate $x$ on $[0,1]$? Here is an attempt: Let $\mathcal{A}$ be the linear span of our set $\{x^0, x^2, x^3, x^4, ...\}$. $\mathcal{A}$ is a vector subspace and separates points. It is also a subalgebra since…
is it normal
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