Mathematics Stack Exchange
Mathematics Stack Exchange

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
3
votes
1 answer

Question about Hilbert-Schmidt Operators

I have already tried but I failed. I can't show it is. I used this way: $| (Kf)_n |^2 \leq c_n \|f\|^2$, and therefore $\|Kf\|^2 \leq \|c\| \|f\|^2$, so that $\|K\| \leq \sqrt{\|c\|} $ is bounded, but I cant take result.
user149385
  • 31
3
votes
0 answers

Non unit version of Stone-Weierstrass theorem

If we assume the Stone-Weierstrass theorem, how to prove the following statement: If $X$ is compact Hausdorff, $ C(X \to \mathbb R)$ is the set of continuous real valued functions. If $ A \subset C(X \to \mathbb R)$ is a closed algebra without the…
user112564
  • 3,552
  • 1
  • 23
  • 51
3
votes
0 answers

Compatible maps and Sub- compatible maps

Definition Let $M$ be a subset of a metric space $X$. The set $M$ is called $q$- starshaped if $ kx+(1-k)q \in M$ for all $x\in M$ and $k\in[0,1]$, where $q$ is an element of $M$. Definition Let $M$ be subset of metric space $X$ and $T, I :M\to M$…
Anil Kumar
  • 53
  • 4
3
votes
1 answer

Densely injective bounded linear map that is not injective

Suppose $T:X\rightarrow Y$ is a continuous linear map of Banach spaces, say. Let $D$ be a dense subspace of $X$ and assume $T$ is injective on $D$. Does it follow that $T$ is injective? I would guess not, but what is a counterexample? Does this…
Jeff
  • 7,139
3
votes
1 answer

$2$-Normed Spaces

Someone suggested today that $2$-normed spaces are actually equivalent to normed spaces. Can anyone who's familiar with the topic provide a counterexample? (I can't access Gähler's original paper introducing the notion, and hence have no way of…
12455421
  • 105
3
votes
1 answer

Proving $\lim_{t\to+\infty}(I + tS)^{-1}f = P_{N(S)}f$ for all $f\in H$ if $S\in \mathcal{L}(H)$ and $(Su, u) \geq 0$ $\forall u\in H$

I'm having trouble filling in the details of a step in one of the proposed solutions to exercise 5.20 from Brezis' functional analysis book. The exercise is as follows ($H$ denotes a Hilbert space with inner product $(\cdot, \cdot )$, while its norm…
Detached Laconian
  • 757
3
votes
1 answer

$c_0$ is a closed subspace of $l^{\infty}$

Put $$ l^{\infty} = \{ (x_n) \subseteq \mathbb{C} : \forall j \; \;\ \;|x_j| \leq C(x)\} $$ I want to show that $c_0$, the space of all sequences of scalars that converges to $0$ is closed subspace of $l^{\infty}$. MY atttemt: Take arbitrary…
user124140
3
votes
5 answers

example of nonexpansive mappings

As we know : $T:X\longrightarrow X$ is a nonexpansive mapping iff $\|Tx-Ty\|\leq\|x-y\|,$ $\forall x,y\in X$ So my question is I want some nonexpansive mappings, I know $\sin(x)$ , $\cos(x)$ and if I'm not wrong $\frac{1}{x}$ for $x\geq1$ Thanks.
Vahid
  • 1,055
3
votes
1 answer

How to construct the Singular Value Decomposition(SVD) of an operator P?

I am reading a paper about SVD of a operator. I know a little about SVD of matrix, and eigenvalue decomposition of operator. But I totally don't know what does SVD of a operator mean. In the following text, could anyone tell me the expressions of…
Yao Jin
  • 91
3
votes
0 answers

A m-dissipative operator is the generator of a $C_0$-semigroup

Let $(H;(\cdot,\cdot))$ be a complex Hilbert space and let $A: D(A) \rightarrow H$ be a densely dened closed operator on H. Let A be an m-dissipative operator that is \begin{align} &(Ax, x) \leq 0 \text{ for all } x \in D(A) \text{ and }…
simon
  • 446
3
votes
1 answer

Example of a separable space without a Schauder basis.

Can I say that the normed linear space $(\Bbb{R}(\Bbb{Q}), \lvert\, \cdot\,\rvert)$ is an infinite dimensional, separable, Banach space and hence cannot have a Schauder basis? My argument is based on the fact that an infinite dimensional Banach…
Alexander
  • 635
3
votes
1 answer

Finding the adjoint for a diagonal operator in $\ell^2$.

Let $\{e_n\}_{n=1}^{\infty}$ be an orthonormal basis of the complex Hilbert space $\ell^2$. Fix complex numbers $\lambda_1,\lambda_2,\lambda_3,\dots$, let $$ \mathscr{D}(T)=\{\sum_{n=1}^{\infty} x_n e_n\in \ell^2:\sum_{n=1}^{\infty}|\lambda_n…
simon
  • 446
3
votes
1 answer

Constructing Unbounded Linear Maps

Suppose that $X$ is an infinite-dimensional normed vector space over $\mathbb C$ (that is, the cardinality of any of its Hamel bases is infinite) and let $Y$ be another, nontrivial normed vector space over $\mathbb C$ (that is, $Y$ contains other…
triple_sec
  • 23,377
3
votes
1 answer

space of continuous function on real line, completeness and separability

Let $C[0,\infty)$ be the space of all continuous real valued functions on $[0,\infty)$ with metric: $\rho (\omega_{1},\omega_{2})=\sum_{n=1}^{\infty}(1/2^{n})\max_{0\leq t\leq n}(|\omega_{1}(t)-\omega_{2}(t)|\wedge 1)$. Show that under $\rho$, the…
swang999
  • 127
3
votes
1 answer

Task - bounded functional operators

Let $r,x\in \mathbb{R}$ , $0
Laki888
  • 133
1 2 3
…
99 100