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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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Compactness in $L(X, Y)$

Let $X$, $Y$ be Banach spaces and $L(X, Y)$ be the space of continuous linear mappings between them equipped with the operator norm. Is there a criterion for when a subset $\mathcal{L} \subseteq L(X, Y)$ is relatively compact? In my textbook, I…
Smiley1000
  • 950
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Functional Analysis (Lax, 2002) p. 75, Theorem 7, Why over $\mathbb{C}$?

Theorem 7. $X$ is a normed linear space over $\mathbb{C}, Y$ a linear subspace of $X$. For any $z$ in $X$, denote by $m(z)$ its distance from $Y$ : $$ m(z)=\inf _{y \in Y}|z-y| $$ We claim that for every $z$ in…
evenzhou
  • 506
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Separable subspace $E_o$ such that $T: E_o\rightarrow F$ is onto.

This is from a 2011 qualifier in functional analysis: Let $T: E\rightarrow F $ be a linear continuous surjective map between Banach spaces. If $F$ is separable, prove there is a separable closed subspace $E_o$ of $E$ such that $T(E_o)=F$. Of…
Kadmos
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Question regarding the proof of the Riesz representation theorem

Every Hilbert space $H$ is isomorphic to its dual space. The map $\Phi: H \rightarrow H', y \mapsto f_{y}$, where $f_{y}:H \rightarrow \mathbb{K}, x \mapsto \langle x,y\rangle$ is a bijective isometric and conjugate linear map. I saw an outline of…
Peter
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A doubt on invertibility of a bounded linear operator

Let $C[0,1]$ be the Banach space of real valued continuous functions on $[0,1]$ equipped with the supremum norm. Define $T: C[0,1] \to C[0,1] $ by $$T(f(x))=\int_{0}^{x}xf(t)dt.$$ Let $R(T)$ denote the range of $T.$ Consider the following…
MathRookie2204
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Finding two extensions of this linear functional

Let X := $\mathbb{C}^{3}$ equipped with the norm $|(x, y, z)|_{1} := |x| + |y| + |z|$ and $Y := \{(x, y, z) ∈ X|x + y = 0, z = 0\}$. Find at least two extensions of $ℓ(x, y, z) := x$ from $Y$ to $X$ which preserve the norm. What if we take $Y :=…
MathGeek
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Mackey-Arens theorem

I'm trying to understand the following theorem of the book Banach space theory of Fabian et. al.: I'm having troubles to see why is it true what I've marked in yellow. As I see it, we have that $U$ is $\mathcal{T}$-closed, convex and balanced. So,…
Eparoh
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Whether for operators that are close enough, the properties of injectivity and surjectivity are kept

Suppose $X$ and $Y$ are Banach spaces, and let $A$ be a bounded operator from $X$ to $Y$. Does there exist $r\gt0$ such that for every $B\in L(X,Y)$ and $\left\|A-B\right\|\lt r$: $A$ is surjective implies $B$ is surjective, or $A$ is injective…
Homer
  • 31
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Is is true that the intersection of two dense subspaces of a linear normed space is also dense?

I tried to find a counterexample for it but I failed. If the statement is true, I think it is sufficient to prove that the intersection of two dense subspaces is also dense in the unit ball, but I don't know how to approach it.
fdaeqw
  • 85
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Dense subset of a dual space norming?

I'm wondering if a dense subset of the dual space is always norming. Precisely, let $E$ be a Banach space. Then for all $e\in E$, $$\|e\|=\sup\limits_{\substack{e'\in E'\\\|e'\|=1}}|e'(e)|.$$ Now assume that $D\subset E'$ is dense. Do we…
Noobtron
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Need help with Exercise 14, Chapter 6 of Rudin's Functional Analysis

The exercise goes: Suppose $K$ is the closed unit ball in $\mathbb{R}^n$, $\Lambda\in\mathcal{D}'(\mathbb{R}^n)$ has its support in K, and $f\in C^\infty(\mathbb{R}^n)$ vanishes on $K$. Prove that $f\Lambda = 0$. Find other sets $K$ for which this…
Bill
  • 33
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Sigma additivity of spectral measure on the complex Borel-sigma algebra

My exact question is asked and answered here: Proving that a spectral measure is $\sigma$-additive but I have spent all day trying to construct the $A^{'}_js$ and $B^{'}_k$s described in the answer and haven't succeeded. I know A und B must each be…
Top Secret
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How to show $L^2(\mathbb{R}, u^{*}TX)$ is a Banach space

I'm very slowly trying to work my way through these notes on obstruction bundle gluing, section 2.1. One exercise I want to do is show that $L^2(\mathbb{R}, u^{*}TX)$ is a Banach space and $L^2_1(\mathbb{R}, u^{*}TX)$ is a Banach manifold. To give…
cheeseboardqueen
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What is a canonical embedding?

I‘m taking a course in functional analysis and have to show that $ L^1(\mathbb R) \subsetneq (L^\infty(\mathbb R))^* $ "in the sense of canonical embedding". What does this mean exactly? Unfortunately, "canonical embedding" is no term we defined in…
ATW
  • 689
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$\lim_{j\xrightarrow{} \infty} |x_j|=0 \Rightarrow [x_j]=[0]$

$x_j$ is a sequence, define its equivalent class as $$ [x_j]=\{(z_j)_{j\geq1} \mid (z_j - x_j)_{j\geq 1}\in Y\}\\ \text{where} \ Y \ \text{is a linear subspace} $$ Do we have the following statement? $$ \lim_{j\xrightarrow{} \infty} |x_j|=0…
K. Zhu
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