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

Closed Convex Hull

Let $X$ be a topological vector space. For a countable subset subset $(a_n)_n\subset X$, can one describe the closed convex hull of $(a_n)_n$ as the set $\{\sum_{n=1}^{\infty}c_na_n: 0\leq c_n\leq 1, \ \text{and} \ \sum_{n=1}^{\infty}c_n=1\}$ where…
user124910
  • 3,007
2
votes
0 answers

Duality pairing in Banach spaces

Suppose we have a Banach space $X$ and its dual $X^{'}$. Then the duality pairing is often written $$\langle f,v\rangle_{X^{'},X} = f(v)$$ for $f \in X^{'}$ and $v \in X$. But is it allowed to write $$ \langle v,f\rangle_{X,X^{'}} $$ to mean…
tgtt
  • 317
2
votes
2 answers

Show that $\varphi \in E'$ and if $E$ is a Banach space then $\varphi \in E$

Problem: Let $E$ be a normed space over field $\mathbb{C}$. Fix a continuous function $f: \left[ a,b \right] \rightarrow E$ with $\left[ a,b \right] \subset \mathbb{R}$. Consider $\varphi: E' \rightarrow \mathbb{C}$ given by $\varphi(y) :=…
Minh
  • 983
2
votes
2 answers

Question about weak topology on normed space

Let $(X,\|\quad\|)$ be a normed vector space, and let $X^\prime$ be the set of all bounded linear maps on $X$. I need help to clarify these questions. Is it correct that the weak topology on $X$ is the topology $\tau_w$ such that for each $f \in…
tgtt
  • 317
2
votes
0 answers

Dual of $(C_\infty(\mathbb{R}^n),\|\cdot\|_\infty)$?

What is the dual space of $X = (C_\infty(\mathbb{R}^n),\|\cdot\|_\infty)$, the continous functions that approach zero at infinity? Is it reflexive? Additionally, is there a Banach space $Y$ such that $X=Y'$?
Jas Ter
  • 1,539
2
votes
1 answer

Functional analysis: locally compact hausdorff space

Let $X$ be a locally compact Hausdorff space. Show that every function in $C_0(X)$ (continuous functions that vanish at infinity) can be arbitrarily uniformly approximated by functions in $C_{00}(X)$ (continuous functions of compact support). In…
CHM
  • 33
  • 4
2
votes
1 answer

a reference for the isomorphism $\ell_1(E')\cong [c_0(E)]'$

Let $E$ be a Banach space, and let $E'$ denote its topological dual. Let us consider the spaces $\ell_1(E')$ and $c_0(E)$ defined by $\ell_1(E')=\{(x_n^{'})_{n=1}^\infty\subset E': \sum_{n=1}^\infty||x_n^{'}||<\infty\}$, and…
serenus
  • 851
2
votes
1 answer

About non-filtering family of seminorms

Let $X$ be a real vector space and let $P$ be a family of seminorms on $X$. We say that $P$ is filtering if for any $p_1,p_2\in P$ we can find $q\in P$ and $c_1,c_2>0$ such that $c_1p_1\le q$ and $c_2p_2\le q$ both hold on $X$. I got no problem of…
Juniven Acapulco
  • 9,654
2
votes
1 answer

Help showing a linear functional is bounded

Let $(X,M,\mu)$ be a $\sigma$-finite measure space, and $k:X\times X \rightarrow \mathbb{C}$ (or $\mathbb{R}$) be $X\times X$ measurable. Suppose there are measurable functions $h,g:X\rightarrow(0,\infty)$ and constants $c_1,c_2>0$ such that…
Scott
  • 719
2
votes
1 answer

How to prove $\vert\bullet\vert\circ\pi^{-1}$ is a norm

Suppose the seminorm on X is real valued, $U=X\bigcap\lbrace x:\vert x\vert =0\rbrace$, Y=X/U, and $\pi:X\rightarrow Y$ is the canonical projection. I want to show that $\vert\bullet\vert\circ\pi^{-1}$ is a norm on Y. But I do not know how to prove…
user627221
2
votes
1 answer

Prove a subspace is separable

Given a Hilbert space $H$ and let $K$ be a compact set in $H$. Let $X$ be the smallest closed subspace containing $K$. Prove that $X$ is separable. Compact implies that $K$ is totally bounded. But how to use this prove $X$ is separable? And how to…
Q-Y
  • 1,589
2
votes
1 answer

Properties of Norm spaces

Suppose $m(E)<\infty$ and $f\in\mathcal{L}^{\infty}(E)$. The goal of this problem is to show \begin{align*} \lim_{p\to \infty}\|f\|_p=\|f\|_{\infty}. \end{align*} First, prove that \begin{align*} \lim_{p\to\infty}\|f\|_p\leq…
user
  • 173
2
votes
0 answers

Condition on $f$ that will imply that $\operatorname{supp}(f*g) = \operatorname{supp}(f)$

$\DeclareMathOperator{\supp}{supp}$ Given $h,k \in L_1(\Bbb R)$ , define $(h*k)(x) = \int_{\Bbb R} h(t)k(x-t)dt$. for any function $h$, define $\;\supp(h) = \{x:h(x)\ne 0\}$. Now ,let $f,g \in L_1(\Bbb R)$. I have showed that $\supp(f*g)\subset…
user335501
2
votes
2 answers

About closed graph of an unbounded operator

I am working on problems related to the closed graph of an unbounded operator. There is a proposition: Let $X,Y$ be Banach spaces and let $A:\mathrm{dom}(A)\to Y$ be linear and defined on a linear subspace $\mathrm{dom}(A)\subset X$. Prove that the…
Apocalypse
  • 888
2
votes
1 answer

$L_d = \{ \phi \in L_{[-a,a]}^2 \vert \phi(t) = -\phi(-t) \}$, prove that is a closed subspace of $L^2_{[-a,a]}$

$L_d = \{ \phi \in L_{[-a,a]}^2$ $\vert$ $\phi(t) = -\phi(-t)$ $\forall t \in [-a,a] \}$, prove that is a closed subspace of $L^2_{[-a,a]}$. Now, the subspace part is always the same and I managed. The part on closeness is where I stumble. This is…
qcc101
  • 1,345
1 2 3
…
99 100