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, ...

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5
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1answer
80 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, ...
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0answers
9 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
79 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
54 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
25 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
17 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
22 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
38 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
15 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 ...
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0answers
39 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
35 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
27 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
32 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, ...
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vote
0answers
24 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
40 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
23 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
32 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
103 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
29 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
53 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
42 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
12 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
29 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
16 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
15 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
22 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
9 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
41 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
46 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
45 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
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 ...
2
votes
1answer
130 views

Algebraic and orthogonal complements

I have been trying to understand this Encyclopedia of mathematics article. Specifically, in the comments section there is the following comment: The codimension of a subspace $L$ of a vector space ...
1
vote
1answer
75 views

Definition of s-lim? (context: Trotter product formula)

I am searching for a definition of "s-lim", a notation I am seeing used sometimes in the statement of the Trotter product formula (for instance in Barry Simon's book Functional Integration and Quantum ...
1
vote
1answer
24 views

Significance of closedness of a subspace when writing a Hilbert space as a direct sum

I read that if $U$ is a closed subspace of a Hilbert space $H$ then we can write $H$ as $H = U \oplus U^\bot$ (the direct sum). What is not clear to me is why $U$ is required to be closed. I thought ...
1
vote
2answers
143 views

How to define orthogonal complement in an arbitrary vector space

In this article about codimension there is the following remark: The codimension of a subspace $L$ of a vector space $V$ is equal to the dimension of any complement of $L$ in $V$, since all ...
5
votes
0answers
72 views

Pointwise approximation of a closed operator

If $T:\mathcal D(T) \rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$, is it possible to find bounded operators $T_n\in \mathscr B(\mathcal ...
0
votes
1answer
36 views

About two subspaces of (1,2)-Sobolev space

I have a question about Sobolev space. Let $\Omega$ be an open subset of $\mathbb{R}^{d}$, we consider the Sobolev space $H^{1}(\Omega):=\left\{ u \in L^{2}(\Omega) : D_{j}u \in L^{2}(\Omega), ...
4
votes
1answer
131 views

Fractional powers of positive self-adjoint operators

Consider two positive unbounded operators $A$ and $B$ densely defined on a Hilbert space $H$ self-adjoint on a domain $\mathcal{D}(A) = \mathcal{D}(B) = H_1$. By the spectral theorem, we can define ...
2
votes
1answer
81 views

Example of Non-separable stochastic process.

This question is related to the link: https://www.encyclopediaofmath.org/index.php/Separable_process The link provided a basic definition of separable Stochastic process. I felt all the process ...
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vote
2answers
559 views

Example of a basis of $C[0,1]$

What are examples of a basis of $C[0,1]$? (Hamel basis or Schauder basis... related: What is the difference between a Hamel basis and a Schauder basis?)
55
votes
3answers
2k views

Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z…: unified treatment of transforms?

I understand "transform methods" as recipes, but beyond this they are a big mystery to me. There are two aspects of them I find bewildering. One is the sheer number of them. Is there a unified ...
2
votes
1answer
51 views

If $u_n^p \rightharpoonup v$ in $L^1$, then does it follow that $u_n \rightharpoonup v^{\frac 1p}$ in $L^p$?

Let $\Omega$ be a bounded domain. Suppose that $u_n^p \rightharpoonup v$ in $L^1(\Omega)$. Does it follow that $u_n \rightharpoonup v^{\frac 1p}$ in $L^p(\Omega)$?