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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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when does the natural pairing correspond to $L^2$ inner product?

When can the natural pairing between a vector space and its dual be thought of as the $L^2$ inner product? I.e. when is there a bijective correspondence between the two? $\textbf{Edit:}$ Let $V$ be a vector space with dual $V^*$. There's a map $V^*…
Stuck
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What is the inner product for the dual Sobolev space?

Let $\dot{H}^{1}(\mathbb{R}^n)$ be the homogeneous Sobolev space formed by completion of $C^{\infty}_{c}(\mathbb{R}^n)$ functions with respect to the norm $||\nabla u||_{L^2}.$ Thus the inner product on $\dot{H}^{1}(\mathbb{R}^n)$ we have, $$\langle…
Student
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Norm attaining functionals on a Banach space

Do every linear functional on a Banach space attains its norm? In case of finite-dimension, the argument follows via compactness of the unit ball. What about in infinite-dimensional case? Is there any any general version of it? Any help will be…
user884919
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Prove: $T: X \rightarrow Y$ with $Tx=x$ is bounded

I want to show that $T: X \rightarrow Y$ with $Tx=x$ is bounded and surjective. where we have $||x||_X = \sum_{n=1}^{\infty}|x_n|$ and $||x||_Y = sup_{n \in \mathbb{N}} |x_n|$ $X:[ {x = (x_1, x_2, x_3, . . .) : x_n ∈ \mathbb{K},…
Mathlover
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Is this product of $L^2$ functions also an $L^2$ function?

Let $I=(l_0,l_1)$ and $g(x)= f(x)\chi_{(\alpha,\beta)}$ where $(\alpha,\beta)\subset I$ and $f \in L^\infty(I)$ with $f(x)\geq c>0$ for all $x\in I$. Now, if $h \in L^2(I)$, is the product $g(x)h(x) \in L^2(I)$ if $f(x)$ is also bounded? My…
CharlesXavier
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A theorem on book Functional Analysis by Brézis

Theorem 2.10, on page 37 Let $E$ be a Banach space. Assume that G and L are two closed linear sub spaces such that $G+L$ is closed. Then there exists a constant $C \geqslant 0$ such that every $z \in G+L$ admits a decomposition of the form $z=x+y$…
MichaelS
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Find the spectrum of $Ax(t)=x(t+s)$

Let be $\mathcal{C}_\mathbb{R}$ the set of the continuous and bounded function in $\mathbb{R}$ with norm $ \|x\|=\underset{t\in\mathbb{R}}{sup}|x(t)|$ Consider the operator $A$ such that $Ax(t)=x(t+s)$ for some $s\in\mathbb{R}$. Show that the…
ifthenelse
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Are there densely defined symmetric operators from a Banach space into its algebraic dual which do not map into the continuous dual?

The notion of symmetric operator on Hilbert space can be generalized to locally convex spaces as follows: A map $T \colon D \to X^a$ between a subspace $D$ of a locally convex topological vector space $X$ and its algebraic dual $X^a$ is called…
n_flanders
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Let $E$ be a Banach space, $F=\mathcal{l}^1$, and let $T\in\mathcal{L}(E,F)$ be surjective, prove that $T$ has a right inverse

this is an exercise from Brezis Book, 2.11. My attempt: T is surjective so thanks to the open mapping theorem, for every $n\in\mathbb{N}$ there exists $x_n\in E$ such that $\|x_n\|\leq c$ ($c$ is the constant from the open mapping theorem) and …
ifthenelse
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Does weak*-convergence imply convergence of the operator norms?

Let $\mathcal A$ be a unital C*-algebra with topological dual $\mathcal A^*$ and denote the unit ball as $B_1^*:=\{\phi \in \mathcal A^* : \vert\vert \phi \vert\vert_{sup}\leq 1\}$. If $\phi_n \rightarrow \phi$ is a weak*-convergent sequence with…
madison54
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On convergence of WOT

Let $H$ Hilbert space. $\{A_n\}_{n\ge0}$ a sequence in $B(H)$ such that the sequence $\{\langle A_nx,y\rangle\}_{n\ge0}$ converges for all $x,y\in H$. Can we show that there exist $A\in B(H)$ such that $$\lim_{n \to \infty}\langle…
Noob mathematician
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A equivalent condition of nonzero linear functional?

** problem.** Let $X$ be an infinite-dimensional normed vector space and let $φ$ be a nonzero linear functional defined in $\,X.\,$ Then the following are equivalent. (i) $φ$ is bounded. (ii) The kernel of $φ$ is a closed linear subspace of…
user158796
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Sturm-Liouville Problem

Possible Duplicate: Sturm-Liouville Problem How could one prove that there are at most countably many eigenvalues of the Sturm-Liouville problem $−Lu=ju$, $j$ = eigenvalue, and $u$ is in $C^2[a,b]$? I have put some more thought and I am still…
Carlotto
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Linear Functional Null Space

I've been having trouble with this problem (2.8, problem 9) from 'Introductory Functional Analysis with Applications' by Kreyszig. "Let $f\neq0$ be any linear functional on a vector space X and $x_0$ any fixed element of $X-\mathbb{N}(f)$, where…
Oleg
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Separable Banach spaces without Schauder basis??

I know that Per Enflo gave examples of separable Banach spaces without Schauder basis, but... I have seen that: 1.- Every vector space has a basis. 2.- A normed space is separable if and only if it has a dense subspace of countably infinite…
Jorge Ortiz
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