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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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Is a contractive automorphism of a Banach algebra an isometry?

If $A$ and $B$ are C*-algebras and $T:A\to B$ is an injective $^*$-homomorphism, then $T$ is necessarily isometric, that is $\|T(a)\|=\|a\|$, for every $a$ in $A$. In particular, every automorphism of $A$ is automatically isometric. However this is…
Arkady Kitover
  • 101
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Reproducing kernel Hilbert space

May I ask a silly question? I want to be sure if it is true that the unions of reproducing kernel of all points in Hilbert space spanned it? i.e. if $K: X \to \mathbb{C}$ is reproducing kernel of H, then $\cup K(.,x) = H \, \, \forall x \in X $?
hala sami
  • 69
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Definition of inner product of Hilbert–Schmidt operator

Wikipedia (Hilbert–Schmidt operator) states the following statement: $ \langle A,B\rangle _{\text{HS}}=\operatorname {Tr} (A^{*}B)=\sum\limits _{i}\langle Ae_{i},Be_{i}\rangle .$ The second equation confuses me a lot. According to my computation,…
Wembley Inter
  • 632
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why isomorphism preserves the function values.

It is known that every functional $g \in l_{q}^{*}(\mathbb{K})$ can be identified with functional $\varphi_{g} \in l_{p}^{**}(\mathbb{K})$ and that every element $y \in l_{q}(\mathbb{K})$ can be identified with a functional $f_{y} \in…
DIEGO R.
  • 1,144
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Difference between spaces $\ell^2$, $L^2$, and $H^2$

Can you please help to understand what exactly the definitions of these spaces, actually I'm so confused. Firstly, the Hilbert space $\ell^2 =\{ \{a_n\}_0^\infty: \sum_0^\infty |a_n|^2 <\infty$ }. Then can I conclude that if $(a_0, a_1, a_2, ...)…
hala sami
  • 69
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On linear transformations

Do there exist continuous linear maps $T\colon V \to V$ such that $T^{-1}\colon V\to V$ exists but is not continuous? Clearly if $V$ has finite dimension the answer is no.
Student
  • 1,687
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Alternative proof for Fredholm's alternative

I am trying to prove the following variant of Fredholm's alternative: Let $X$ be a Banach space, $T \in GL(X)$ (invertible) and $A \in K(X)$ (compact operator). Prove that $T+A$ is invertible iff $T+A$ is injective. The invertible $\implies$…
Anon
  • 1,757
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Does weak* convergence together with convergence of norms implies strong convergence in l1 as dual of c0?

Let $x^{*}_{n}$ is weak$^*$ convergent to $x^*$ in weak$^*$ topology on $l_{1}$ induced by $c_{0}$ and $\|x^{*}_{n}\|\rightarrow \|x^{*}\|.$ It is true that then we have $\|x^{*}_{n}-x^{*}\|\rightarrow 0$?
rose93
  • 21
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Finding image and kernel of $ T:PC[0,1]\to PC[0,1]$ such that: $Tf(x)=\frac{1}{\sqrt{2}}f\big(\frac{x}{2}\big) $

I think I might be completely off on the following question, so would appreciate your opinions and advice: Given: $$ T:PC[0,1]\to PC[0,1]$$ $$ Tf(x)=\frac{1}{\sqrt{2}}f\big(\frac{x}{2}\big) $$ (where $PC$ stands for piecewise-continuous functions) I…
Anon
  • 1,757
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Linear homeomorphisms mapping an orthonormal basis into another orthonormal basis

Consider $L^2(A)$ and $L^2(B)$. If $\{a_i\}$ is an o.n basis of $L^2(A)$, how many linear homeomorphisms $F:L^2(A) \to L^2(B)$ do there exist such that $Fa_i$ is an orthonormal basis of $L^2(B)$? Is this a very restrictive assumption on the maps,…
aere
  • 967
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How to prove that the operator is linear continuous and find its norm?

Let $A:X\to Y$, where $X=C^1[0,1]$ equipped with the norm $\|f\|_{C^1}=\|f\|_u+\|f'\|_u$, and $Y=C[0,1]$ equipped with the uniform norm $\|f\|_u$ be defined as $$(Af)(x)=f'(x);$$ I proved the linearity as…
Andrij Matviiv
  • 33
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Linear operator not bounded is not compact

Let $B$ a banach space infinite dimensional, let X a normed space, and $T: B \to X$ a linear operator such that $\|T(x)\|_{X}\geq c \|x\|_{B}$ for all $x\in B$ and $c>0$ Then $T$ is not compact This theorem perhaps extends the fact that a…
wessi
  • 195
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Convergent subsequence under image of adjoint.

Let $X$ and $Y$ be normed spaces, and let $T : X \rightarrow Y $ be a bounded linear operator, and denote by $T^*$ the adjoint operator of $T$. Suppose also that $X$ is separable, that is, $X$ admits a countable dense subset. Let $ (y_n)_{n \in…
turnip_man
  • 141
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Rewrtiting the $L^p$-norm with $\mu(\{|f|>t\})$

I am looking at an exercise (6.2) in Kesavan's book on Functional Analysis. It says: Let $(X, S, \mu)$ be a measure space and let $1 \leq p < \infty$. Define, for $t>0$, $$ h_f(t) = \mu(\{|f| > t\}. $$ Show that $$ ||f||_p^p = p \int_0^\infty…
Fred Kel
  • 23
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Weak convergence of a sequence of characteristic functions in general

Given $c>0$, can we produce sequence of measurable sets $E_n\subset\mathbb{R}$ such that their characteristic functions $\chi_{E_n}$ converge weakly in $L^p(\mathbb{R})$ to some $c\chi_A$ for some $A\subset\mathbb{R}$ with…
user999577
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