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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).

2
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2answers
45 views

Is the norm $||x||:= \sum\limits_{i=1}^{\infty} 2^{-i}|x_i|$ equivalent to usual norm in $l_2$?

Is the norm $$||x||:= \sum_{i=1}^{\infty} 2^{-i}|x_i|$$ equivalent to usual norm in $l_2$? I have already shown that it is a norm, I expect that it is equivalent, but I can not prove it.
1
vote
1answer
25 views

Distance to set

Let $S$ be a non empty set in an inner product space $E$ Show that if $x\in E,z\in E$ and $Re<x-z, y-z> \le0$ for each $y\in S$ then $d(x,S)=||x-z||$ I would like a clue on how to approach ...
0
votes
0answers
16 views

structure of subrepresentations of (infinite) sums of irr. representations

Let $G$ be a (locally compact) group and $ ( \pi_1 , V_{\pi_1} ) , ( \pi_2 , V_{\pi_2} ) , \ldots$ irreducible, unitary representations. I think, I can show, that for the (finite) direct sum ...
1
vote
1answer
24 views

How do I interpret the boundedness in this space?

As I was reading the article Non linear elliptic and parabolic equations involving measure data, I came across with the following: the sequence $\{f_n \}$ is bounded in the space $L^1(0,T;W^{-1,...
0
votes
0answers
48 views

A question on linear integral equation about non degenerated bilinear form

Let $X$ be a Banach Space , $X\subseteq H,\bar{X}=H$,where $H$ is a Hilbert space $i:=X\to H$ defined by $i(x)=x$ and is continuous. Define $\langle x,y\rangle=\langle ix,iy\rangle$ then $\langle X,X,...
1
vote
2answers
39 views

Weak continuity of the addition and scalar multiplication

Let $X$ be an infinite dimensional normed vector space. Show that vector addition and scalar multiplication are weakly continuous. $$+:X×X \rightarrow X; +(x,y)=x+y$$ $$•:\mathbb {R}×X \rightarrow X; •...
1
vote
1answer
35 views

Uniform Boundedness of Sequence

Let $c_{0}=\{(x_{i})_{i\geq 1} | x_{i}\in\mathbb{K}\text{ and } \lim_{i\rightarrow\infty}x_{i}=0\}$. Suppose $y_{i}\in\mathbb{K}$, $i=1,2,\ldots$, such that \begin{align} \sum_{i=1}^{\infty}x_{i}y_{i},...
0
votes
1answer
19 views

Equivalent definitions of extreme point

We know that a point $x$ of a convex set $K$ in a normed linear space $X$ is said to be an extreme point if for any $y,z\in K$ and for any $0<\lambda<1$, $$x=\lambda y+(1-\lambda)z,$$ then $x=y=...
1
vote
0answers
15 views

Eigenvalue of algebraic multiplicity $m$ is a pole of the resolvent of order $m$.

Let $X$ be a Banach space and $T \in \mathcal{L}(X)$ be a bounded linear operator. Suppose that for some isolated point $\lambda \in \sigma(T)$ and some $m \in \mathbb{N}$ we have $\ker(T-\lambda I)^m ...
-1
votes
1answer
16 views

Why does the determinant vanishing imply nontrivial solutions for a set of differential equations?

Why does this set of equations have to be singular? If the determinant doesn't vanish, what does this imply? I'm a mathmatically inclined materials science student, so I'll probably understand ...
1
vote
0answers
34 views

Showing $f(x)$ $=$ $\sqrt{x}\sin(\frac{1}{x})$ satisfies a Holder Condition of $\alpha < 1$

I'm learning about functions that satisfy Holder's Condition of order $\alpha$. Specifically, A function $f$ is said to satisfy a Hölder condition of order $\alpha > 0$ if there exist $M$ such ...
3
votes
3answers
88 views

A monotonic function that intersect with all lines in $\mathbb R^2$

Let $f:\mathbb R\to\mathbb R$ be a monotone function. Let $\gamma=\{(x,y)\ |\ y=f(x)\}$ is a curve in $\mathbb R^2$. Does there exists a $f$ such that $\gamma\cap L\neq \emptyset \ \forall L\...
1
vote
0answers
43 views

Why do we need projection in the definition of the Stokes operator?

$\DeclareMathOperator{\div}{div}$ $\def\bu{\mathbf{u}}$ Let $D$ be the square $[0,1]^2$ and consider the following space: $$ V:=\{\bu: \bu\in H^2(D)^2, \div \bu=0, u|_{\partial D}=0 \}. $$ Introduce ...
0
votes
0answers
37 views

Is it possible for a sequence of matrices to have pointwise but not uniform convergence?

Is it possible for a sequence of matrices to have pointwise but no uniform convergence? The norm for the matrices is the operator norm.
0
votes
0answers
30 views

Show that Lipchitz functions space is Banach

Let X be a Banach space. Show that $L = \{ f:X \to \mathbb{R}: f \text{ is lipschitz and } f(0)=0 \}$ with norm: $$||f||_{lip}= sup \left\{ \frac{|f(x)-f(y)|}{||x-y||}; x \neq y \in X \right\}$$ ...
0
votes
1answer
22 views

Determining if $f \in BV([0,1])$.

I am reading about Riemann-Stieltjes Integration in Carother's Real Analysis. We find that functions of Bounded Variation provide us with a rich class of integrators. Therefore, I am trying to learn ...
0
votes
1answer
26 views

Weak convergence to 0 iff bounded and pointwise convergence to 0

I am working on a problem from functional analysis that has me stumped. Let $B$ be a reflexive Banach space on some subset of $\mathbb{R}^n$ s.t. point evaluations are continuous. Show that if $f_n\...
2
votes
1answer
48 views

Show that space is not metrizable

Let $X=C([0,1], \mathbb{R})$, $f \in X$, $\epsilon >0$ and $x_1,...,x_n$ in [0,1]. Consider: $V_{f,x_1,...,x_n,\epsilon}= \{ g \in X : |f(x_i)-g(x_i)| < \epsilon, i=1,...,n \}$ and $\tau= \{ ...
3
votes
0answers
37 views

Every Banach limit on $l_{\mathbb{C}}^{\infty}(\mathbb{N})$ is an extension of some Banach limit on $l_{\mathbb{R}}^{\infty}(\mathbb{N})$

Let $l_{\mathbb{C}}^{\infty}(\mathbb{N})$ be the space of bounded complex-valued sequences, $l_{\mathbb{R}}^{\infty}(\mathbb{N})$ the subspace of real-valued sequences. Given any Banach limit $L_1: l_{...
2
votes
1answer
47 views

Convergent sequence of functions and compact set contains the spectrum of the limit

Let $K\subseteq \mathbb{C}$ be a compact subset. Let $(x_n)$ be a convergent sequence of normal elements in a unital $C^*$-algebra $\mathcal{A}$, with limit $x$, such that $\sigma(x_n)\subseteq K$ ...
2
votes
0answers
44 views

If $\kappa$ is a contractive operator on $C_0(ℝ)$, is $λ(κ-\text{id})$ the generator of a Feller semigroup?

Let $E$ be a locally Hausdorff space, $$C_0(E):=\left\{f\in C(E):\left\{|f|\ge\varepsilon\right\}\text{ is compact for all }\varepsilon>0\right\},$$ $\kappa$ be a Markov kernel on $(E,\mathcal B(E))...
0
votes
0answers
52 views

Is the integral equal to zero?

Consider $$\int_{S_r} \left(\frac{\partial\phi}{\partial\eta}\right)^2 (x\cdot\eta)\ d\sigma $$ Where $S_r$ is the sphere of radius $r$ and centered at zero, $\phi\in L^2(\mathbb{R}^n)$ and $\eta$ is ...
1
vote
2answers
36 views

Sufficiency in the proof that $L^p(\mu)$ is complete

In the proof that $L^p(\mu)$ is complete for $p\in[1,\infty]$ (as done in Saxe, Theorem 3.21 or in Folland, Theorem 6.6, the latter of which is outlined here) we make use of the following completeness ...
0
votes
1answer
56 views

Does the series corresponding to a Cauchy sequence **always** converge absolutely?

Let $X$ be a normed vector space and consider a Cauchy sequence $(x_n)_{n\in\mathbb{N}}$ in $X$. Is it true that the corresponding series of our Cauchy sequence, $\sum_{i=1}^\infty x_i$, always ...
1
vote
1answer
60 views

Hilbert Transform: limit of xHf(x)

In Terence Tao's notes page 1, cited below, he mentions that it is easy to see that $\lim_{|x| \to \infty} xHf(x) = \frac{1}{\pi}\int f$ where $f$ is a Schwartz function and $H$ is the Hilbert ...
0
votes
0answers
13 views

What is the defining properties of the heat kernel?

According to https://en.wikipedia.org/wiki/Fundamental_solution, the fundamental solution $u$ is defined as the solution of $Pu = \delta$. In the meantime, we know that the fundamental solution of the ...
1
vote
1answer
54 views

Riemann integrability and discontinuity

$g:[0,1] \rightarrow \mathbb{R}$ bounded and $\alpha:[0,1] \rightarrow \mathbb{R}$ non-decreasing. Assume $ g \in \mathbb{R}_\alpha[\delta,1]$ for every $\delta > 0$. I showed that $g\in\mathbb{...
1
vote
1answer
23 views

Convergence of finite dimensional projection of trace class in trace norm

Assume $\mathbb{H}$ is a Hilbert space and $K$ is a trace-class operator on it. Given a fixed ONB $\{e_i\}$ and assume $$K=\sum_{i,j}c_{ij}e_i\otimes e_j.$$ Now, let $K_n = \sum_{1\leq i,j\leq n}c_{...
3
votes
1answer
59 views

An optimization problem in $L^1(0,1)$

Is there any non-negative function $f(t)$ that minimizes $\int_0^1e^{\int_0^tf(s)ds}dt$ and satisfies $\int_0^1sf(s)ds =1$? I guess there is not, because the exponential is minimized if $\int_0^tf(s)...
0
votes
0answers
12 views

The inclusion of Sobolev spaces is compact

I know that the inclusion of Sobolev spaces with compact support is a compact map. Now I wonder whether the inclusion of isotropic Sobolev spaces is compact. My definition of the isotropic Sobolev ...
4
votes
0answers
97 views

If $f$ is integrable, then $\| f\|$ is also integrable.

As usual, a partition of a compact interval $[a, b]$ is, by definition, an strictly increasing family $\Pi = (t_k)_{k = 0}^m$ ($m \geq 0$) of points in the interval such that $t_0 = a$ and $t_m = b;$ $...
2
votes
0answers
97 views

Nonexistance result of elliptic equation

I want to prove that there is no $L^2(\mathbb{R}^N)$-solution of the equation $−\Delta \phi =\lambda \phi$ for every $\lambda \in \mathbb{R}$. I know that the Pohozaev identity asserts for $N\geq 3$, $...
0
votes
1answer
20 views

Closure of Continuously DifferentiableFunctions in Holder Space

Here is a curious (already submitted) homework problem I had in analysis some time ago: Let $\Omega$ be a convex domain in $\mathbb{R}^n$ with $C^1$ boundary. Let $C^{0,\alpha}(\overline{\Omega})$ ...
2
votes
0answers
20 views

If $A \in \mathcal{L}_c(X)$ and $X$ is Banach, then $\dim \ker (\text{id}-A) < + \infty$.

Exercise : Let $X$ be a Banach space and $A \in \mathcal{L}_c(X)$. Show that $\dim \ker ( \text{id} - A) < + \infty$. Attempt/Thoughts : The kernel of the operator $(\text{id}-A) : X \to X$ ...
8
votes
1answer
72 views

Heat semigroup norm between fractional Sobolev and $L^p$ spaces

What is the actual inequality that holds for the heat semigroup between fractional Sobolev space $W^{2\alpha,p}$ and classical Lebesgue space $L^q$? I am trying to derive an inequality $$ \lvert\...
1
vote
2answers
25 views

If $A \in \mathcal{L}(H)$ and $\langle A(u),u \rangle \geq \langle u, u \rangle$, then $A$ is invertible.

Exercise : Let $H$ be a Hilbert space and $A \in \mathcal{L}(H)$ such that : $$\langle A(u),u \rangle \geq \langle u, u \rangle \; \forall u \in H$$ Show that $A$ is invertible. Attempt/...
4
votes
1answer
38 views

Are the bounded Lipschitz functions dense in $L^1(\mu)$?

I am currently reading a paper (L. Ambrosio and B. Kirchheim. Currents in metric spaces) and I stumbled uppon a fact which I don't know how to prove. I have the following setting: Let $X$ be a ...
0
votes
0answers
25 views

Show that graph of operator with adjoint operator is closed

Let $X,Y$ be inner-product spaces. Let $T\in L\left(X,Y\right)$ be a linear operator with adjoint operator $S\in L\left(Y,X\right)$ such that $$\langle Tx,y\rangle_Y=\langle x,Sy\rangle_X\quad\forall (...
2
votes
0answers
23 views

Why is this operator symmetric? A question concerning a paper from Brezis and Crandall

I am reading the paper Uniqueness of the initial-value problem $u_t - \Delta \varphi u = 0$ by Brezis and Crandall. At the beginning of the analysis, they introduce an operator $B g$ by $$ B g := (\...
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vote
0answers
25 views

Density of a subspace

Let $\Omega\subset\mathbb{R}^d$ be open, bounded and simply connected with smooth boundary $\partial\Omega$. Define $\mathcal{H}:=\{u\in H^1(\Omega)~|~\Delta u\in L^2(\Omega)\}$ with norm $$\|u\|_\...
2
votes
1answer
39 views

Several questions about bilinear forms on Banach spaces

I have been reading Fundamentals of Differential Geometry by Serge Lang. Let me briefly describe his definitions. In the following, bilinear forms are always assumed to be continuous. A Banach space $...
1
vote
0answers
31 views

Lagrangian density of second order hyperbolic equation

It's been some hours now that I am trying to find the Lagrangian density of the following hyperbolic PDE with variable coefficients and $c_{ij}=c_{ji}(x)$ $$ \partial^2_t u - c_{ij}\partial_i\...
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vote
0answers
22 views

Spectrum of an operator defined by spectral integral

First of all I want to thank you for the help you provide on this website! Whenever I had a hard time understanding things in math I visited this website and (nearly) allways found a hint or a ...
3
votes
0answers
12 views

Convergence in $W^{1,\infty}$ implying convergence in $W^{-1,\infty}$.

I'm looking for references or proofs of the following facts: Weak star convergence in $L^\infty$ implies strong convergence in $W^{-1,\infty}_{\text{loc}}$, and Strong convergence in $L^\infty$ + ...
2
votes
0answers
47 views

Proving that a spectral measure is additive

Given the medible space $(\mathbb{R},\mathbb{B(\mathbb{R})})$ (where the latter is the borel's $\sigma$-algebra), a Hilbert space H and two spectral measures $E,F:\mathbb{B(\mathbb{R})}\rightarrow B(H)...
2
votes
1answer
42 views

Show that there is a continuous function $g$ on $[-\pi,\pi]$ satisfying $||f-g||_{2} < \epsilon$

I'm learning about Fourier Series from Carother's Real Analysis. I'm ultimately trying to prove the following result: Let f $\in$ $R([-\pi,\pi])$, then ||$f$ - $S_{n}(f)||_{2}$ $\overset{n \to \...
2
votes
1answer
44 views

Convergence of Indicator function in weak*-topology

Let $\omega_n$ and $\omega$ be measurable subsets of $[0,1]$. Also let indicator function $\chi_{\omega_n}$ converge to $\chi_{\omega}$ in weak*-topology on $L^\infty(0,1)$, that is $$\int_0^1f(x)(\...
1
vote
1answer
43 views

Weak-* continuous functionals with bounded level sets

Let $X$ be an infinite-dimensional Banach space, $X^*$ its topological dual and $f:X^*\to\mathbb{R}$ some weak-* continuous functional (not necessarily linear). Is it possible for $f^{-1}(a)$ to ...
1
vote
1answer
30 views

Riemann's Lemma as a corollary of Bessel's Inequality

I am learning about Fourier Series with Carother's Real Analysis. I'm currently stuck on the bolded statement below made by the author. First I will give a little context. We have just proved the ...
0
votes
0answers
24 views

Bijection+Closability?

I was wondering whether a closable operator $A$ with domain $D(A)\subset H$ ($H$ is a Hilbert space) which is also bijective has an everywhere defined bounded inverse? Thanks in advance. Math.