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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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Understanding the definition of the generator of a semigroup of operators

Given a strongly continuous semigroup $T : \mathbb{R}_+ \to L(B)$ on a Banach space $B$, its infinitesimal generator $A$ of a strongly continuous semigroup $T$ is defined as a mapping $B \to B$ as $$ A\,x = \lim_{t\downarrow0} \frac1t\,(T(t)-…
Tim
  • 47,382
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Subspace of $L^2(U)$ that are closed under multiplication?

Suppose $U \subset \mathbb{R}^n$ is non-empty, bounded, open, connected with $C^1$ boundary. Is there any subspace of $L^2(U)$ that are closed under multiplication, that is under which conditions can we say that $$ u ,v \in L^2(U) \implies uv \in…
Henrik Finsberg
  • 1,387
2
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2 answers

Is every operator adjoint for something?

Let $X$ and $Y$ be a Banach spaces and let $S\in B(Y^\ast,X^\ast)$. Do we always have $S=T^\ast$ for some $T\in B(X,Y)$? I know that every bounded linear operator has an adjoint, also bounded linear operator. But in this case I don’t even know…
thing
  • 1,690
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Computation of Dixmier trace

Let $s$ with $Re(s)> 1$ and let $d>1$ be an integer. Consider the series $$ f(s) = \sum_{k=0}^\infty \frac{\binom{k + d -1}{k}}{(1+k)^{ds}} $$ I would like to show that the residue of $f$ at $s = 1$, i.e. $\lim_{s \to 1^+} (s-1)f(s)$, is $1/ d!$.…
k76u4vkweek547v7
  • 1,314
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Prove $p(\alpha_n x_n)\to 0$ provided $\{\alpha_n\}_{n\in\mathbb{N}}$ converges and $p(x_n)\to 0$

Let $E$ be a vector space. A map $p : E \to \mathbb{R}$ satisfies : $p(x+y)\leq p(x)+p(y)$ ; For a fixed $x\in E$, $\lambda \mapsto p(\lambda x)$ is continuous ; For all $\lambda\in\mathbb{R}$, if $p(x_n)\to 0$ as $n\to+\infty$, then $p(\lambda…
ling
  • 1,589
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Find a linearly independent set of bounded linear functionals $\{f_1, \dots, f_n\}$ such that $f_i(x) = \alpha_i$ where $\alpha \in \Bbb R^n$

From Functional Analysis by Kreyszig: If a linear operator $T:X \rightarrow Y$ on a normed space has a finite dimensional range. show that $T$ has a representation of the form $$Tx = \sum_{i=1}^n f_i(x)y_i$$ where $\{y_1, \dots, y_n\}$ and $\{f_1,…
Oliver G
  • 4,792
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Counter-example to the Banach–Steinhaus theorem (uniform boundedness principle)

I am trying to figure out why completeness is necessary in this theorem. And I was given the following task: Consider $C[0,1]$ with norm $$\|x\| =\int_{0}^{1}| x(t)| dt$$ and operators $$A_nx = n\int_{0}^{1/n} x(t) dt$$ So I want to prove that this…
whatanhonor
  • 183
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Provide an example of $X$ and $ A$ where $\inf_{y\in A} \Vert x-y \Vert <1$ for every $x \in X $ with $\Vert x \Vert =1$

Let $X$ be an inner product space, and $A$ a closed subspace of $X$ with $A\neq X$. Provide an example of $X$ and $ A$ where $\inf_{y\in A} \Vert x-y \Vert <1$ for every $x \in X $ with $\Vert x \Vert =1$ My attempt: I try to start with $A=\{ f\in…
Flashhh
  • 409
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Extreme points of the closed unit ball in space of convergent sequences.

Consider the Banach space $c$ of convergent sequences over $\mathbb{C}$ with the infinity norm defined by $\lvert\lvert(x_n)_{n\in\mathbb{N}}\rvert\rvert_\infty = \operatorname{sup}(\{x_n\in\mathbb{C}:n\in\mathbb{N}\})$. I am trying to characterize…
Jarne Renders
  • 987
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If $f(x_\lambda) \to f(x)$ for all $f \in C(K)$, then $x_\lambda \to x$ in $K$

Let $K$ be a Hausdorff compact set and take a net $(x_\lambda)$ and a point $x$ in $K$. Suppose that $f(x_\lambda)$ converges to $f(x)$ for all $f \in C(K, \mathbb R)$. How can one proof that $x_\lambda \to x$? In this problem, I'm considering…
user 242964
  • 1,898
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1 answer

Graph of a function $f$, around a point $x_0$, with $f'(x_0)<0$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function. Let $x_0$ be some point in which $f$ has derivative s.t $f'(x_0)<0$. How does the function graph looks around $x_0$ Well, in my opinion, it is only the second graph, because it is…
naruto25
  • 461
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1 answer

Characterization of unitarily equivalent direct sum of irreducible operators

Let $(\mathscr{H_i})_{i\in I}$ be a collection Hilbert spaces and define $\mathscr{H} = \{h:I\rightarrow \cup_i\mathscr{H}_i:h(i)\in \mathscr{H}_i,\ \sum_{i\in I}\|h(i)\|^2<\infty\}$. Then it is easy to show that $\mathscr{H}$ is a Hilbert space.…
OgvRubin
  • 1,371
2
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2 answers

Showing an operator is compact

Can someone please explain to me how I can show the following operator is compact? $$ Af(x)=\int_0^xf(t)dt$$ where $A:L_2[0,1]\rightarrow L_2[0,1]$. There are two definitions of compactness that I've encountered: Def1. An operator $L$ is compact if…
Teodorism
  • 1,165
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Does UU* is a projection in Hilbert space iff U*U is a projection?

Suppose $H$ is a Hilbert space. Let $U:H\rightarrow H$ be a bounded linear operator. Does $U^{*}U$ is a projection implies $UU^{*}$ is a projection? Of course, once we prove this, it follows that one of them is a projection is equivalent to another…
Harry Check
  • 135
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1 answer

On the existence of a basis with good properties in finite-dimensional normed spaces

Consider a proper non-trivial subspace $M$ of a finite-dimensional normed space $X$. We can always find a basis $\mathcal B=\mathcal B_1 \cup \mathcal B_2$ for $X$ such that $\mathcal B_1$ is a basis for $M$. However, I wonder whether or not we can…
André Porto
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