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, as well as measure, ...

learn more… | top users | synonyms (1)

2
votes
1answer
538 views

Sequence of measurable functions converging a.e. to a measurable function?

I understand if $(X, \Sigma, \mu)$ is a measure space, and we have a sequence of measurable functions $f_{n}$ such that $\lim \limits_{n \to \infty} f_{n}$ exists almost everywhere d$\mu$ (a.e. ...
0
votes
1answer
16 views

Prove that $(a_n) \in l_2$.

Suppose that $\sum_{k=1}^{\infty} a_k x_k< \infty$ for all $x=(x_n)\in l_2$. Prove that $(a_n) \in l_2$. My attempt Let $T_n: l_2 \to \mathbb{K}$, $T_n(x) = \sum_{k=1}^{n} a_k x_k$. ($\mathbb{K} ...
2
votes
1answer
23 views

Show that finite dimensional subspace is closed

We know that if $V$ is a normed vector space and $W$ is a finite dimensional subspace of $V$, then $W$ is closed. One way to prove this is to show that $W$ is actually complete. Since complete space ...
10
votes
0answers
304 views

Norm Inequality on a Compact Riemannian Manifold

Consider the following problem: Suppose $(M, g_{ij})$ is a compact Riemannian manifold. Assume $u$ is a smooth, nonnegative function which satisfies the differential inequality $$\Delta u \geq ...
5
votes
2answers
41 views

For a normal operator is it true that $\|T^*T^2\| = \|T^3\|$?

For a normal operator is it always true that $\|T^*T^2\| = \|T^3\|$? See the accepted answer for the case in a Hilbert space Update: how about $\|T^*T^2\| = \|T\|^3$ in a Hilbert space
2
votes
1answer
51 views

Projective sequence of C*Algebras by factors of embedded ideals isomorphic to algebra

Let $A$ be a $C^*$-algebra and $$A = I_1 \supset I_2 \supset I_3 \supset\ldots$$ be a sequence of embedded ideals in $A$ such that $\bigcap_{i=1}^\infty I_i = 0$. Is it true, that the projective limit ...
3
votes
0answers
28 views

Compact sets of compact-open topology

Let $X$ and $Y$ be topological spaces,$X$ not compact, denote with $C(X,Y)$ the set of continuous functions between $X$ and $Y$ and put on $C(X,Y)$ the compact-open topology. My question is: which ...
0
votes
1answer
11 views

Linearspan of Gaussians dense in Schwartz space

as the title already says I am trying to show that the linear span "A" of the gaussians $e^{\frac{-|x|^2}{2}}$ and their translations/ dilations are dense in the Schwartzspace. This is the space of ...
0
votes
1answer
26 views

Prove that $(a_n) \in l_\infty$

Suppose that $\sum_{k=1}^{\infty} a_k x_k< \infty$ for all $x=(x_n)\in l_1$. Prove that $(a_n) \in l_\infty$ My attempt I tried to apply Uniform boundedness principle to the linear functionals ...
2
votes
0answers
16 views

Necessery and sufficient condition for existence of star cyclic vector for $M_\phi$

Let $X$ be a $\sigma$-finite measure space. $M_\phi :L_2(\mu)\rightarrow L_2(\mu)$ for $\phi \in L_\infty (\mu)$ is defined by $f \rightarrow \phi. f$. $f_0$ is called a star cyclic vector for ...
5
votes
1answer
77 views

$f_1,…,f_n$ be linear functionals on a real vector space $V$, then is there a norm on $V$ which makes every $f_i$ continuous?

Let $V$ be a real vector space, $f_1,...,f_n$ be linear functionals on $V$; then does there exist a norm on $V$ with respect to which each of $f_i$ is continuous? And what if we have infinitely many, ...
0
votes
0answers
5 views

Question on weighted Sobolev spaces

Let us define a weighted Sobolev space $W^{k,p}_\delta(\Omega)$ as \begin{equation} W^{k,p}_\delta(\Omega) = \left \{ u \in L^p(\Omega): (1+r^2)^{\frac{1}{2}(-\delta-\frac{3}{p}+|\beta|)}D^{\beta}u ...
2
votes
2answers
77 views

Definition of Dirac Delta function on the surface of a unit sphere

I am looking for a definition of a Dirac Delta function which is defined on the 2D unit sphere surface in 3D. In other words, I am looking for a function which is zero everywhere on the 2D spherical ...
0
votes
2answers
53 views

Generalization of Cauchy-Schwarz to positive operators

The problem I am given is: Let $T$ be a positive operator. Prove that for all $x,y$ we have $$|\langle Tx,y\rangle| \le \langle Tx,x\rangle^{\frac{1}{2}} \langle Ty,y\rangle^{\frac{1}{2}}.$$ ...
1
vote
1answer
18 views

Problem about a compact operator $T:l^p\rightarrow l^p$

I have to solve this problem. Let $\{\lambda_n\}$ be a sequence of real number such that $\lim_{n\rightarrow\infty}\lambda_n=0$ and consider the operator $T:l^p\rightarrow l^p$, $1\leq p\leq ...
3
votes
1answer
12 views

Poisson Equation: 'Dirichlet' type problem on all of $\mathbb{R}^N$

I've seen a great deal about solving the Dirichlet problem for Poisson's equation \begin{equation} \Delta u = f \quad \mbox{in} \quad \Omega \subset \mathbb{R}^N\\ u = 0 \quad \mbox{on} \quad ...
1
vote
1answer
21 views

Use contraction mapping theorem to show that the integral equation has a unique continuous solution on $t \in [0,3]$

I have to use the contraction mapping theorem to prove that the integral equation with continuous functions $K(t,s)$ and $f(t)$, $\begin{equation*} x(t) = \lambda \int_{0}^{3} K(t,s) x(s)ds + f(t), ...
1
vote
1answer
30 views

calculate solution of heat equation with method of separation of variables

let the following problem $$ \begin{cases} \dfrac{\partial u}{\partial t}= \dfrac{\partial^2 u}{\partial x^2}, 0<x<1, t>0\\ u(0,t)=0, u(1,t)+ \dfrac{\partial u}{\partial x}(1,t)=0\\ ...
1
vote
0answers
10 views

Fourier Sequence Converges Uniformly Implies Almost Everywhere Pointwise Convergence

I'm trying to understand this problem: Let $f$ be Riemann integrable on $[0,2\pi]$ Suppose that the Fourier Series of $f$, $S_{n}^{f}(x)$, converges uniformly on the interval. I want to show that ...
1
vote
0answers
38 views

What is the norm of the dual space $H^1(\Omega)'$?

I am working on the Bidomain-Model which, during a time interval [0,T], describes the electrical behaviour of the myocardial muscle considered as $\Omega \subset \mathbb{R}^3$. This model has partial ...
4
votes
1answer
25 views

The second isomorphism theorem for C*-Algebras

in my functional analysis class right now we are studying the basics of C* Algebras and I was recently asked this question about the second isomorphism theorem for C* Algebras, but first let me cite ...
1
vote
0answers
24 views

Sobolev spaces on Riemannian manifold

I know one can define the Sobolev spaces on a Riemannian manifold as completions, but is there an equivalent definition that uses weak derivatives, like in the case of open sets in $\mathbb{R}^n$ ? ...
1
vote
1answer
18 views

Singular integral operator

i got the following problem to solve. Let $0 < \alpha < 1$, $L \in L_\infty([0,1]^2)$, $D = \{(x,y) \in \mathbb{R}^2: x = y\}$ the diaagonal of $\mathbb{R}^2$ and $k:[0,1]^2 \setminus D \to ...
0
votes
1answer
29 views

Characterization of elements of $X^*$ via the Radon-Nikodym theorem

I am reading Lindenstrauss' Classical Banach Spaces II and I am having trouble with the following characterization of integrals. First a couple of preliminary definitions: Let $(\Omega, \Sigma, ...
1
vote
0answers
21 views

If Fourier sequence $\{H_n\}$ of a function $f$ converges almost everywhere to a function $g$, then $f=g$ almost everywhere?

if $\{H_n\}$ fourier sequence of a function $f$ converges almost everywhere to a function $g$ then $f=g$ almost everywhere Where $H_n = n$th fourier series of $f$. True or False?
2
votes
2answers
38 views

heat equation-uniqueness of solution

let the following problem $$ \begin{cases} \dfrac{\partial u}{\partial t}= \dfrac{\partial^2 u}{\partial x^2}, 0<x<1, t>0\\ u(0,t)=0, u(1,t)+ \dfrac{\partial u}{\partial x}(1,t)=0\\ ...
0
votes
0answers
17 views

If the continuous action of a non-compact Banach-Lie group on a Banach space preserves the zero element, then it is non-proper.

I am studying the differential geometry of Banach-Lie groups, specifically, the differential geometry of the orbits of an action of a Banach-Lie group on a Banach space, and I ended up "proving" the ...
-1
votes
1answer
18 views

Sequence converging to a closed set

How can I prove that A sequence generated by a particular algorithm converges to a set? i.e irrespective of the starting point the sequence converges to any of the points in a set, or oscillates ...
1
vote
1answer
31 views

Schwartz function whose Fourier transform is compactly supported and $\geq 1$ on the unit ball.

I need to construct such a function but the closest I have come to is to take $f(t) = e^{-|t|}, t\in\Bbb{R^d}$. But its Fourier transform is not compactly supported as is $\hat{f}(x) = ...
3
votes
2answers
99 views

Show $F_b (\Omega, X)$ is a Banach space

Let $\Omega$ be any non empty set and let $X$ be a Banach space over $\mathbb{C}$. Let $F_b (\Omega,X)$ be a linear subspace of $F(\Omega, X)$ of all functions $f; \Omega \to X$ such that ...
0
votes
0answers
28 views

For any function $f$ in $L^2(-π,π)$, is it true that $||f||_{L^2}$ $\leq$ $C||f||_{\infty}$?

For any function f in $L^2(-\pi,\pi)$, is it true that $||f||_{L^2}$ $\leq$ $C||f||_{inf}$ ? I came up with this question because in An Introduction to Hilbert Space by N.Young, right before ...
5
votes
0answers
52 views

Compactum of Banach algebra

I need an example of Banach algebra $A$ and a left non-trivial closed ideal $I$ with all of following properties: There exists a bounded approximate identity in $I$ for $I$ i.e., a net ...
1
vote
1answer
33 views

Rellich's theorem fails at the end point.

If $\{f_k\}\subseteq H^s(\Bbb{R}^d)$ is a bounded sequence and support of each $f_k$ is contained in a common compact set, then there exists a subsequence that converges in $L^q$ for $\forall$ $q, ...
0
votes
0answers
9 views

Norm vs A-norm in non-Archimedean Functional Analysis

Let $K =(K,| \cdot |)$ be a non-Archimedean valued field. Let $E$ be a $K$-vector space. A norm on $E$ is a map $||\cdot||:E\to[0,\infty)$ such that: $||x||=0$ if and only if $x=0$, $||\lambda ...
0
votes
0answers
25 views

Does convergence almost everywhere to an $L^p$ function and existence of a weakly convergent subsequence guarantee weak convergence?

By assumption, for $p \in (1,\infty)$, I have a bounded sequence of functions $f_n$ in $L^p$ (that is, $L^p$ norms of the functions are uniformly bounded) that converges almost everywhere to a ...
0
votes
0answers
15 views

Bounded sequence in Sobolev space has a convergent subsequence

Suppose $\{\phi_k\}$ are a bounded sequence of distributions on in $H^s(\Bbb{R^d})$, where $0<s<\frac{d}{2}$ and $H^s$ is the Sobolev space. Then the sequence has a subsequence that converges in ...
1
vote
0answers
14 views

Root distance function in Metric space [duplicate]

Let $\mathbf X = \Bbb R$ with distance function defined by $d(x,y) = {|x-y|}^\alpha$ , where $\alpha \in \Bbb R$ $(0<\alpha\le1)$. Prove that $(\Bbb R , d)$ is a metric space. The first three ...
0
votes
0answers
19 views

Spectral radius of an element in a Banach algebra

I want to solve this problem: Let $A$ be an unital Banach algebra and $a\in A$ then \begin{equation} r(a)<1 \Longrightarrow \lim_{n\to \infty} a^n=0 \Longrightarrow (1-a)^{-1}=\sum_{n=0}^\infty ...
0
votes
0answers
9 views

Compostion of tempered distribution and linear map.

While solving a particular problem about composition of tempered distributions and an affine transformation, I ended up having to prove the following for $u\in\mathscr{S}'$ and a linear transformation ...
0
votes
0answers
8 views

Extremum of Laplacian of a function

Let f(x,y,z) be any arbitrary continuous function. Let's denote Laplacian of f by $\nabla^2 f$. 1) How do we denote the extremum of $\nabla^2 f$ mathematically ? 2) how do we solve for such ...
0
votes
1answer
32 views

state of a C* algebra

If $A$ is unitial algebra and if $\omega$ is a state of $A$, then for all $a\in A$ $$|\omega (a)|\leq\omega (\left | a \right |^{2})^{\frac{1}{2}}$$ How can I prove this corollary? I would like to ...
1
vote
1answer
40 views

Norm of linear operator

Given two real numbers $\alpha$ and $\beta$, consider the linear operator $T:\mathbb{C}\rightarrow \mathbb{R}$ defined by $T(x+iy)=\alpha x +\beta y$. I am trying to figure out the norm of this ...
1
vote
3answers
44 views

Show $Px \perp (x-Px)$ and prove $||P|| \leq 1$

Let $H$ be a Hilbert space and linear operator $P:H \rightarrow H$ satisfy $\langle Px,y\rangle = \langle x,Py\rangle$ for all $x,y \in H$ and $P^2=P$. (a) Show that $Px \perp (x-Px)$ for every $x ...
3
votes
1answer
40 views

Continuity of functional calculus

Let $\mathcal{A}$ be an unital C*-Algebra. $a,b$ be normal elements in $\mathcal{A}$. $X\subset \Bbb C$ is a compact subset. $f:X\rightarrow \Bbb C$ is continuous. I need to show that for all ...
-1
votes
0answers
24 views

Regularity of the solutions of the infinite dimensional dynamical systems

Consider a densely defined unbounded operator $A:D(A)(\subset H)\to H$ which is infinitesimal generator of a strongly continuous semigroup $\mathbb{T_{t\ge0}}$ for the following dynamical system: ...
1
vote
2answers
56 views

Closed subspace of a functional space and its orthogonal complement

Let's consider the following subspace of $C[-1, 1]$ with the following inner product $(f, g) = \int_{-1}^{1}{f(t) \overline{g(t)} dt}$: $$H_{0} = \{ f \in H | \int_{-1}^{0}{f(t) dt} = ...
1
vote
1answer
34 views

Cauchy-Schwarz inequality for dual pairing?

Suppose we have a Hilbert space $X$ and its dual $X^*$. Given a dual pairing $$_{X^*}\langle x,y\rangle_X,$$ does there exist a sort of Cauchy-Schwarz inequality so that $|\langle x, y\rangle|\leq ...
1
vote
1answer
38 views

Fredholm index - Motivation behind it.

I have a question concerning the motivation behind the Fredholm index: What is it good for? I know that there are many theorems dealing with it, for example that it is continuous, invariant under ...
0
votes
1answer
61 views

Is there a relation between Ill-posed problems and Eigenvectors.

One can easily explain an ill-posed problem with an equation AX=b. The following link is an good example: https://www.encyclopediaofmath.org/index.php/Ill-posed_problems 1) Can there be a class of ...
3
votes
1answer
64 views

Is the 3d Schwartz space isomorphic to a subspace of the 1d Schwartz space?

Are the Schwartz-spaces $\mathscr{S}(\mathbb{R})$ and $\mathscr{S}(\mathbb{R}^3)$ isomorphic (as topological vector spaces)? Is $\mathscr{S}(\mathbb{R}^3)$ at least isomorphic to a subspace of ...