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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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How to show that a operator has finite rank?

Let $E,F$ be normed vector spaces. Consider the mapping $\sum_{k=1}^n y_i \otimes f_i: x \mapsto \sum_{k=1}^nf_i(x)y_i $, where $x \in E, y_i \in F, f_i \in E'$. Show that the mapping above is a continous linear operator with finite dimensional…
wanymose
  • 562
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Why is the simple energy functional $\int_\Omega |\nabla u|^2 $ weakly sequentially lower semi-continuous?

I want to prove the existence and unicity of a minimiser of the energy functional $u \to E(u) = \int_\Omega |\nabla u|^2$ for $\Omega$ a smooth domain in $\mathbb{R}^n$ with $C^1$ boundary and $u \in U := \{ u \in H^1(\Omega) : u|_{\partial \Omega}…
Suspicious Fred
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$N$ be a closed vector subspace of a vector space $L$

I am not able to do the following , could anyone help me? $N$ be a closed vector subspace of a vector space $L$ such that $L/N$ finite dimensional, we need to show that any subspace of $L$ containing $N$ is closed. Thank you.
Myshkin
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On recovering a bounded operator from a bi-linear form

Let $\mathcal{H}$ be real Hilbert space, $b(\cdot,\cdot)$ is a bi-linear form on $\mathcal{H}$ and satisfies \begin{equation} \begin{split} \sup_{||v||=1}\sup_{||u||=1}b(u,v)\leq C_0\cdots(1)\\ \inf_{||v||=1}\sup_{||u||=1}b(u,v)\geq…
Roy Han
  • 911
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$g$ has Hahn-Banach Extension to $\mathbb{R}^2$ uniquely?

$\mathbb{R}^2$ with sup-norm consider $Y=\{(y_1,y_2):y_1+y_2=0\}$, $g:Y\to\mathbb{R}$ is defined by $g(y_1,y_2)=y_2$, I want to know whether $1.$ $g$ has Hahn-Banach Extension to $\mathbb{R}^2$ uniquely? $2$. every linear functional that satisfies…
Myshkin
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Showing that the functional dot product must be positive.

The Question: Given $Lu=-u''$ with $D(L)=\{u(x)|u \in C^2 [0,1], u(0)=0, u(1)=u'(1)\},$ show that $\langle Lu,u \rangle \geq 0$ for all $u \in D(L)$ where $$ \langle f,g \rangle =\int_{0}^{1} f(x)g(x) dx. $$ My attempt: I did integration by parts to…
user546380
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Weak convergence in $L^2$ according to $\alpha$

Let $E=L^2(\mathbb{R})$ endowed with its usual topology. Let $\alpha>0$ and $\left(\varphi_n\right)_{n \in N}\subset E$ defined by $$ \varphi_n=\frac{n^{3 / 2}}{n^{2 \alpha}+x^2} $$ Give, with justification, the values of $\alpha$ for which the…
ali h
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$u(E)\subset v(E) \Rightarrow$ the boundedness of $\|y\|/\|x\|$

Assume $E$ is Banach space and $u,v\in\mathcal{B}(E)$. Prove that if $u(E)\subset v(E)$, then there exists some constant $k\geq 0$ such that for all $x\in E$, there exists $y\in E$ such that $\|y\|\leq k\|x\|$ and $u(x) = v(y)$. Since $u(E)\subset…
Stephen
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Set of Injective Bounded Linear Operators with Closed Image is Open Subset of $L(X,Y)$ wrt Norm Topology

Working on this question for exam preparation and am stumped. Clearly we will want to use the closed image theorem but I am stuck on how to proceed and would appreciate a hint. I am having issues with working with the linear operators rather than…
ConfusedGradStudent
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problem on self-adjoint and A bounded

Problem: Let $A$ be self adjoint and $B$ symmetric. Suppose $B$ is $A$ bounded with bound $\alpha$. Prove that $$ \lim _{n \rightarrow \infty}\left\|B(A+i n)^{-1}\right\| \leq \alpha $$ Definition: Let $\mathscr{H}$ be a Hilbert space, $A: D(A)…
neveryield
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Showing that nested closed sets contains a unit disc.

Let $X$ be a Banach space and $B_n \subseteq B_{n+1}$ be closed sets (with non-empty interior) sets such that $$ X = \bigcup_{n=1}^\infty B_n $$ Is it true that $\{ x\in X: \|x\|=1 \} \subseteq B_N$ for some $N\in \mathbb N$? Edit 1. This is to…
Hash Nuke
  • 535
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Extending derivative to $L^2$

I have written in my notes that the differential operator $Tf(x) = f'(x)$ defined on $C^1[0,1]$ cannot be extended to a closed operator on all of $L^2[0,1]$, although I'm forgetting why. Is it just that if this were the case, then by the closed…
beeclu
  • 304
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non convergent sequence under continuous linear functional on $l_2$

Does the sequence $\{e_i\}$ converges in $l_2$? does the sequence $\{f(e_i)\}$ converges for any continuous linear functional on $l_2$? I know that in $l_2$ norm $\{e_i\}$ does not converges as $d(l_i,l_j)=\sqrt{2}$ always! But I dont know the…
Myshkin
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Why is there an asymmetry between the left and right annihilators?

Let $X$ be a normed vector space, and $X'$ be the dual space of $X$, consisting of all bounded linear functionals on $X$. If $K\subset X$ and $L\subset X'$, then we may define the left and right annihilators to be: $$^\perp L=\{x\in X:\langle…
Void
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Finding the dimension of $\ker A$ and $\mathrm{coker} A$ when $A$ is the differential operator

My problem is the following: All functions are assumed to be real-valued. Determine $\dim(\ker A)$ and $\dim(\mathrm{coker}A)$ when $A = \frac{d}{dx} :X \to Y$ and $X = \{u ∈ C^1([0,1]): u(0) = u(1), u′(0) = u′(1)\},~~ Y = \{f ∈ C^0([0,1]): f(0) = f…
marg
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