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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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Invertibility of compact operators in infinite-dimensional Banach spaces

Let $X$ be an infinite-dimensional Banach space, and $T$ a compact operator from $X$ to $X$. Why must $0$ then be a spectral value for $T$? I believe this is equivalent to saying that $T$ is not bijective, but I am not sure how to show that…
user1736
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What facts about the weak topology fail in spaces that aren't Banach?

I'm learning about the weak and weak* topologies on a normed vector space following the book of Brezis. He limits his discussion to case where $E$ is a Banach space, and my question is most simply stated as, "Why?". I can't find an example of a…
NKS
  • 4,392
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Engineer: Unable to appreciate importance of L2 space

Can someone provide me with some material to make clear of the importance of $L^2$ space in engineering/physics? Having a background in all the introductory mathematics courses offered in engineering, I find myself completely unable to appreciate…
Fraïssé
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Operator whose spectrum is given compact set

Let $A\subset \mathbb{C}$ be a compact subset. Since $A$ is compact and metric space, it is separable, say $\overline{\lbrace a_n\rbrace_{n=1}^\infty}=A$. Let $\mathcal{l}^2(\mathbb{Z})$ be the Hilbert space consisting of $L^2$-summable sequences…
user8484
  • 801
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Does pointwise convergence of continuous functions on a compact set to a continuous limit imply uniform convergence on that set?

The problem is in Marsden - Elementary Classical Analysis 2nd ed. Ch.5. My approach was Condition : $f_k, f(\in\mathcal C) : A\to N$, $f_k \to f \ \text{(pointwise)}$ and $A$ is compact. Since $A$ is compact, the range of $f_k, f$ are all compact,…
Jinmu You
  • 701
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Prove that $C^1[0,1]$ is not complete with norms

I have to show that the $C^1[0,1]$ is not complete with any of these norms: $\|f\|_{\infty}=\sup_{x\in[0,1]}|f(x)|$ $\|f\|_{*}=|f(0)|+\int_0^1|f'(x)|dx$ My attempt The right sequence for the first norm is $f_n=\sqrt{x+\frac{1}{n}}$. Notice that…
luka5z
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Can all continuous linear operators on a function space be represented using integrals?

A linear functional $\omega:v\mapsto\omega(v)$ on a finite dimensional vector space $X$ of dimension $N$ with an inner product $(·\ ,·)$ is an element of the dual vector space $X'$, and a couple of isomorphisms later I'll find…
Nikolaj-K
  • 12,249
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3 answers

Closed subspace of $l^\infty$

I've got here this exercise that says: "Show that $c$ is a closed subspace of $l^{\infty}$" (with $c$ I mean the sequences of $l^{\infty}$ that converge in $l^{\infty}$, with respect to the norm of $l^{\infty}$). I've done it, but I cannot say if it…
batman
  • 2,065
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2 answers

Norm of a tensor product of operators

I have two Hilbert spaces $H_1$ and $H_2$ which are subspaces of a bigger Hilbert space $H$. I also have two bounded linear functions $T_1:H_1\rightarrow H$ and $T_2:H_2\rightarrow H$. I define the tensor product space $F=H_1\otimes H_2$, and a…
Max
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Motivation behind the definition of Banach-Mazur Distance

We know that any two n-dimensional normed vector spaces $X,Y$ are isomorphic, and we define the Banach-Mazur distance between $X,Y$ as $$ d(X,Y)=\inf \{ \|T\|\|T^{-1}\|:T\in GL(X,Y) \} ,$$ where $GL(X,Y)$ is the space of all linear isomorphisms.…
Nirakar Neo
  • 889
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What is the difference between weak and strong convergence?

What is the difference between strong and weak convergence? I am reading "Introductory functional analysis" by Kreyszig and I dont appreciate the differences between the two. Definition of strong convergence: A sequence $(x_n)$ in a normed space…
user232183
  • 293
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$f$ is discontinuous $\iff$ kernel($f$) is dense in X

Problem: Let $X$ be a normed space and $f$ be a non-zero linear functional on $X$. Then prove that $f$ is discontinuous $\iff$ kernel($f$) is dense in $X$. I have proved that if $f$ is discontinuous, then kernel($f$) is dense in $X$. How to…
user265328
14
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1 answer

Finite-dimensional subspaces of normed vector spaces are direct summands

Here is a problem in functional analysis from Folland's book: If $\mathcal{M}$ is a finite-dimensional subspace of a normed vector space $\mathcal{X}$, then there is a closed subspace $\mathcal{N}$ such that $\mathcal{M}\cap \mathcal{N} = 0$ and…
user24367
  • 1,286
13
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1 answer

Norm of adjoint operator in Hilbert space

Suppose $H$ is a Hilbert space and let $T \in B(H,H)$ where in our notation $B(H,H)$ denotes the set of all linear continuous operators $H \rightarrow H$. We defined the adjoint of $T$ as the unique $T^* \in B(H,H)$ such that $\langle Tx,y \rangle =…
caligula
  • 435
13
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1 answer

Closed unit ball is compact?

Let $X$ be a Banach space and let $\operatorname{Lip}_{0}(X)$ be the space of all real-valued Lipschitz functions which vanish at $0$. The space $\operatorname{Lip}_{0}(X)$ is a Banach space when it is equipped with the Lipschitz norm, defined…
Serges
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