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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 contraction idempotent operator self-adjoint?

Is a contraction idempotent operator self-adjoint? In the other words, if $T:H\to H$ is a bounded linear operator such that $||T||\leq1$ and $T^{2}=T$, can we conclude $T=T^*$?
morapi
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Is a functional derivative a generalized function?

I am just now learning about elementary distribution theory, and it seems that theory may bear on the topic of functional differentiation, which I've encountered in some books on quantum field theory (QFT). I'm not looking for mathematical rigor,…
Greg Grunberg
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If $T: X \to Y$ is norm-norm continuous then it is weak-weak continuous

Let $X,Y$ be normed linear spaces (or Banach spaces if necessary) and let $T: X \to Y$ be linear. We call $T$ norm-norm continuous if $X,Y$ are endowed with the norm topology and similarly, weak-weak continuous if $X,Y$ are endowed with the weak…
user167889
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Dense subspace of linear space and functional equal to $0$

Let $X$ be a normed space over field $\mathbb{K}$ and let $X_0$ be a linear subspace of $X$. I have to prove that: $X=\overline X_0 \iff$ For every linear continuous functional $\phi : X\rightarrow \mathbb{K}$ we have $\phi |_{X_0} \equiv0 \implies…
luka5z
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Unbounded linear functional maps every open ball to $\mathbb{R}$?

I can't get my head wrapped around this: Let $X$ be a normed linear space. Let $f:X\rightarrow\mathbb{R}$ be a linear functional on $X$. Prove that $f$ is unbounded if and only if $\forall y\in X$ and $\forall \delta>0$ we have…
Laars Helenius
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Spectrum of unbounded Operators + Spectral Theorem

I've seen a variaty of slightly different definitions for the spectrum and its division into pure point spectrum residual spectrum and so on. Thus I'm wondering what could be an appropriate definition. Moreover, when does the spectral theorem apply…
C-star-W-star
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How should I think about reflexive spaces? What "property" do I get from a space being reflexive?

Let $(X,\| \cdot \|_X)$ be a $\mathbb{R}$-vector space with bidual space $X^{**}$. We defined $X$ to be reflexive, if the canonical embedding $\mathcal I: X \to X^{**}$ with $$\mathcal I x(l) := l(x)$$ for all $l \in X^*, x \in X$ is surjective. I…
Huy
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Applying Open Mapping Theorem

Let $X$ and $Y$ Banach spaces and $F: X \to Y$ a linear, continuous and surjective mapping. Show that if $K$ is a compact subset of $Y$ then there exists an $L$, a compact subset of $X$ such that $F(L)= K$. I know by the Open Mapping Theorem that…
wit
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$\operatorname{Ran} \lambda I - T$ is closed for compact operator $T$ and $\lambda \neq 0$

Let $T$ be a compact operator on a Hilbert space $\mathcal{H}$ and $\lambda \in \Bbb{C} - \{0\}$. I want to show that $\operatorname{ran} \lambda I - T$ is closed. So suppose we have $g_j = (\lambda I - T)f_j \in \operatorname{ran} \lambda I - T$…
user38268
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Excercise 1.13 in Brezis's Functional Analysis

This is the Excercise 1.13 in Brezis's Functional Analysis Let $E=\mathbb{R}^n$ and let $$P=\{x\in\mathbb{R}^n;x_i\geq 0\ \forall i=1,2,...,n\}$$ Let $M$ be a linear subspace of $E$ such that $M\cap P=\{0\}$. Prove that there is some hyperplane…
Danielsen
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Fitting an infinite collection of balls in an infinite dimensional unit ball

Given an infinite dimensional normed linear space, how would one show that it is possible to fit an infinite collection of non-overlapping balls of radius $\frac{1}{4}$ in the unit ball? I guess one can immediately reduce the problem to a normed…
user1736
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What is an example of a bounded, discontinuous linear operator between topological vector spaces?

I am thinking there might be an example between the space of compactly supported smooth functions on the real line (chosen because it is non-metrizable under the standard topology for this space of test functions) and $L^{1/2}[0,1]$ (chosen because…
Wayne
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Extensions of Bounded Linear Operators

I am currently lacking in some basic knowledge regarding extensions of linear operators. Let $ X $ and $ Y $ be Banach spaces and let $ A $ be dense subspace of $ X $. Let $ T : A \to Y $ be a bounded linear operator. Is it true that there…
LMW
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Isometries of $\ell^p_n(\mathbb{C})$

Let $1
user8305
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