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
2
votes
1 answer

Positive Definite Multiplication operator

Let ($X$,$A$,$\mu$) be a $\sigma$-finite measure space and $g$ $\in$ $L^{\infty}(\mu)$. Define $M_g$: $L^2(\mu)$ $\longrightarrow$ $L^2(\mu)$ by setting $M_g(f)$ = $fg$. Show that if $M_g$ a positive definite operator then $g$ $\geq$ $0$ a.e. on…
Ester
  • 3,057
  • 2
  • 23
  • 34
2
votes
0 answers

Riesz theorem without orthogonal projection theorem

Let $H$ be a Hilbert space. I am trying to prove the Riesz theorem, but without using the orthogonal projection theorem. Let $f: H \longrightarrow K$ be a functional. The key is that since $f$ is continuous, $\ker f$ is closed and we can use the…
Minkowski
  • 1,518
2
votes
2 answers

Closed respect a norm $\|\cdot\|$

What mean that expression "closed respect a norm". For example: $$H^{m,p}(\Omega)= \overline{\left\{{u\in C^m(\Omega); \|u\|_{W^{m,p}}<\infty}\right\}}^{\|\cdot\|}$$
user46060
2
votes
2 answers

Infinite intersection of $L^p$ space

Consider the space $$L^{2-}(\mathbb{R}):=\bigcap_{1\leq p<2}L^{p}(\mathbb{R})$$ equipped with the norm $$\Vert f\Vert_{L^{2-}(\mathbb{R})}:=\sup_{1\leq p<2}\Vert f\Vert_{L^{p}(\mathbb{R})}.$$ Is it true that…
Capublanca
  • 1,262
2
votes
1 answer

Generalisation of fundamental theorem of linear algebra

Does something similar to "Fundamental theorem of linear algebra" hold for infinite dimensional Hilbert spaces? https://en.wikipedia.org/wiki/Fundamental_theorem_of_linear_algebra
user415535
2
votes
1 answer

Hahn Banach separation theorem for weak* closed set

I wonder if Hahn Banach theorem still holds on a weak* closed set. To be specific, let $X$ be a Banach space, and $X^*$ is its dual space. Let $S$ be a convex, weak* closed set in $X^*$, and $f\in X^*\backslash S$. Then, is there any point $y\in X$…
ForestM
  • 69
2
votes
1 answer

What are $\|\partial _\nu v_n\|_{H^{-1/2}}(\partial \Omega )$ and $\|v_n\|_{H^{1/2}(\partial \Omega )}$?

We have for example here and here that $$\|v_n\|_{H^1}=1,\|f_n\|_{L^2},\|v_n\|_{H^{1/2}(\partial \Omega )},\|\partial _\nu v_n\|_{H^{-1/2}(\partial \Omega )}\to 0.$$ Q1) I don't really know what mean $\|v_n\|_{H^{1/2}(\partial \Omega )}$ and…
user349449
  • 1,577
2
votes
1 answer

$\mathbb N$ queen

A 8-queen problem is to find a function s.t. $\ f : \{1, 2, 3, 4,5, 6, 7, 8\} \rightarrow \{1, 2, 3, 4,5, 6, 7, 8\}$ $\ f$ bijective $\ f-\mathrm{Id}, \ f+\mathrm{Id}$ injective If I modify the first limitation to $f : \mathbb N \rightarrow…
Mudream
  • 534
2
votes
0 answers

Functional's Distance from Kernel

Let $X=\{f\in C[0,1] ; f(0)=0\}$, $M=\{f\in X ; \int_{t=0}^{1}f(t)dt=1\}$. I have to prove that $\forall f\in X , ||f||_{\infty}=1$ exists $d(f,M)=|\int_{t=0}^{1}f(t)dt|$. I proved $d(f,M)\geq |\int_{t=0}^{1}f(t)dt|$, but I don't know how to prove…
user
  • 244
2
votes
1 answer

Volterra Operator Gives Zero?

Consider the operator $K(\varphi)(t)=\int_{s=-\pi}^{\pi}\sin(t-s)\varphi(s)ds$. If we will look at $K(e^{int})$ we will get zeros $ \forall n\in N$, thus it means that we got the zero operator? (because this is a basis to $L^2[-\pi,\pi]$)
user
  • 244
2
votes
0 answers

riemann stieltjes integrablity

Let $f$ be riemann stieltjes integrable w.r.t $a$; let $g$ be a continious linear functional on $\mathscr{C}[a,b]$. Then $g(f)= \int_{a}^{b} f da$. Is it riesz thorem? Can anyone refer any link to the proof of the theorem? Any help would be…
Math is beautiful
  • 427
2
votes
1 answer

equivalent norms in Sobolev space

In Remark 11 on page 214 ( Functional analysis, Sobolev spaces and partial differential equations written by Haim Brezis): Let $I$ be an bounded interval, let $1 \le p \le \infty,$ and $1 \le q \le \infty.$ From Theorem 8.2 and (5), it can be shown…
04170706
  • 359
2
votes
0 answers

Functionals on the set of coninuous functions with period 1

$C([0,1]_o)$ is the set of continuous functions $f:\mathbb R \rightarrow \mathbb C$ with period 1, with the inifinity norm. I proved that if $g_1 ,g_2$ are functionals in $C([0,1]_o)^*$ that satify $g_1 (e^{2 \pi i n x})=g_2 (e^{2 \pi i n x})$ for…
MasterJ
  • 1,036
2
votes
0 answers

Need for reference, functional analysis

I have some questions that do interest me, but professor does not want to discuss them due to them being hard. That is why I ask for reference here. What is the dual of $L^{+\infty}(X, \mu)$, e.g. the space of all bounded functions in given…
nikola
  • 1,045
2
votes
1 answer

Countable intersection of dense open subsets in complete metric space

Let $X$ be a complete metric space. Let $E_n$ be a nowhere dense subset of $X$ for every $n$. Let $M_n$ be a dense open subset of $X$ for every $n$. Show that $\bigcap_n M_n$ is not contained in $\bigcup_n E_n$. My attempt: Since $X$ is a complete…
user430647
  • 393
1 2 3
…
99 100