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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3
votes
1answer
281 views

Dense subspace of Zygmund space or Hölder space?

Do we know any function spaces dense in Zygmund space $C_*^s$(a special case of Besov space, i.e. $C_*^s = B^s_{\infty,\infty}$) or Hölder space$C^{k,r}$, with underlying field $\mathbb{R}^d$? Will ...
0
votes
1answer
59 views

Prove that the Besov Space is a Banach space

Help me prove that the Besov space is a Banach space. I need to show that the Besov space is complete. If the Besov space is a closed subset of $L_p$ and since all $L_p$ spaces are complete then I'm ...
6
votes
1answer
277 views

Besov spaces---concrete description of spatial inhomogeneity

Some very pedestrian questions about Besov spaces. Just to fix notation: 1.Let $f \in \mathcal{S}'$, the space of tempered distributions. 2.$\Psi, \{ \Phi_n \}_{n \geq 0} \subset \mathcal{S}$ such ...
0
votes
2answers
25 views

A relation between the domain of $A$ and the domain of $\bar A$

Let $A$ be an operator: $$ A:D(A)\to R(A) $$ where $D(A)$ and $R(A)$ are respectively the domain and the range of $A$ and they are subspaces of a Hilbert spcae $(H,\|\|)$. Suppose that $A$ is a ...
1
vote
1answer
20 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
0answers
24 views

Let $H$ be a Hilbert Space with$\langle \cdot,\cdot \rangle$ and $E_1=\{w\in H : Pw=w\}$, show $E_1$ is closed.

Let $H$ be a Hilbert Space with $\langle \cdot,\cdot \rangle$ and $E_1=\{w\in H : Pw=w\}$ with $P:H\rightarrow H$ is linear, $P^2=P$ and $\langle Px,y \rangle=\langle x,Py \rangle \forall x,y\in H$. ...
0
votes
1answer
12 views

Why does a Hermitian operator with singleton spectrum have to be scalar?

One proof of Schur's lemma proceeds by showing that a Hermitian intertwining operator of an irreducible representation (of a topological group on a Hilbert space) has a spectrum that contains only one ...
1
vote
0answers
28 views

Strichartz estimates for wave equations

Let's consider the wave equation $\Box u=F $ with $u(0)=g_0$ and $\partial_t g(0)=g_1$. Strichartz estimates tell us that $$\Vert u\Vert_{L^q_tL^r_x}+\Vert u\Vert_{C^0_t\dot{H}_x^s}+\Vert\partial_t ...
1
vote
1answer
34 views

Show if $x\in l^p$ and $y\in l^q$ with $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ then $\|xy\|_r\leq\|x\|_p\|y\|_q$

Show if $x\in l^p$ and $y\in l^q$ with $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ then $\|xy\|_r\leq\|x\|_p\|y\|_q$. My intuition is to use Young's Inequality and then apply it to $A_k=\frac{|x_k|}{\|x\|}$ ...
3
votes
0answers
21 views

Riesz Lemma with $\alpha=1$ and Linear Bounded Functional

Show that on a normed linear space $X$, Riesz lemma with $\alpha=1 $ holds implies that every bounded linear functional attains its norm on the unit sphere of $X$. This is not a homework question and ...
-1
votes
0answers
21 views

If F is upper semi continuous closed set- valued map

If $F$ is upper semi continuous closed set-valued map from $X$ to $Y$ and $A⊂X$ and $X$ be a complete metric space. The image of the set $A$ under $F$ is given by $$F(A) = \lbrace y: y \in F(x): x ...
1
vote
0answers
10 views

If F is a closed set- valued map from X to Y and A is a subset of X.

If $F$ is a closed set- valued map from $X$ to $Y$ and $A \subset X$, then the image of the set $A$ under $F$ is given by $F(A) = \lbrace y: y \in F(x): x \in A \rbrace $, let $CB(X)$ denote space ...
4
votes
1answer
150 views

Scale invariance and the Mellin transform?

The Mellin transform of a function is given by: $$\mathcal{M}[f](s) = \int_0^{\infty}x^{s-1}f(x)dx$$ Supposedly, the magnitude of the Mellin transform is invariant to scaling, analogous to how the ...
0
votes
0answers
17 views

If a mapping on complete metric space is continuous, then mapping on space of closed and bounded sets of that metric space is also continuous ??

If a mapping on complete metric space $X$ is continuous, then mapping defined for sets, on space of closed and bounded sets of $X$ is also continuous ??
2
votes
1answer
50 views

Calculating the distance function and projection

Let $\Omega=\{x\in\mathbb R^n\mid\langle a,x\rangle=b\}$. We define the distance function and projection as follows $$d(x;\Omega)=\inf\{||x-\omega||\mid\omega\in\Omega\}$$ ...
3
votes
0answers
23 views

Resolvent estimate of hyperbolic Laplacian

Consider the Laplacian $-\Delta$ on the hyperbolic space $\mathbb{H}^n$. For $\lambda \in \mathbb{C} \setminus [0, \infty)$, do we have resolvent estimates of the form $$\Vert (-\Delta - \lambda ...
1
vote
1answer
23 views

Can we conclude that $v_{n}\rightarrow v$ in $L^{\infty}\left(\Omega\right)$ if $p>N$

Let $\Omega\subset\mathbb{R}^{N}$ be a smooth bounded domain, $v_{n}\rightharpoonup v$ in $W_{0}^{1,p}\left(\Omega\right)$ , $\left\Vert v_{n}\right\Vert _{W_{0}^{1,p}}=1$ $\forall n$ . So we ...
-1
votes
0answers
18 views

an objective question from functional analysis [on hold]

Let $A$ and $B$ be bounded operators on a Hilbert space $H$ such that $AB=BA$. Let $\lambda$ be an eigenvalue for $A$. Then it must be that a)$B$ has no eigenvalue b)$B$ has at least one ...
0
votes
1answer
24 views

Laplacian operator on $L^2(\Omega)$?

Let $\Omega\subseteq \mathbb R^n$ be an open subset and $\displaystyle \Delta:=-\sum_{j=1}^n D_j^2$ be the Laplacian operator. I have some questions concearning this operator: $(i)$ Does it map ...
0
votes
0answers
23 views

Young's inequality for convolutions for functions of bounded support

If $$f\in L^P(\mathbb{R}^d), g\in L^q(\mathbb{R^d}), \; \frac{1}{p}+\frac{1}{q}=1+\frac{1}{r},$$ then Young's inequality for convolutions states $$\|f*g\|_{L^r}\leq\|f\|_{L^p} \|g\|_{L^q}.$$ In ...
6
votes
2answers
30 views

Prove Operator is a Projector

Let $\mathscr{H}$ be a complex Hilbert space. A projector is a linear map $P:\mathscr{H}\to\mathscr{H}$ such that $P\circ P = P$. I'm trying to prove the following claim, from the information given ...
1
vote
0answers
21 views

Completing Karlin's proof of variation diminishing transformation theorem

In S Karlin's book total positivity there's a theorem that says if $K(x,y)$ is $TP_r$ (totally positive with degree $r$) and the sign change count of function $h$, $S(h) = n\leq r-1$, then ...
1
vote
0answers
25 views

functional analysis. Compact operator. Hilbert-Schmidt theorem.

I have the following problem: "Under which $ \alpha \in \mathbb{R}$ is the operator $ T: L_2 [1, + \infty) \to L_2 [1, + \infty) $: \begin{equation*} (Tf) (x) = x^{\alpha} \int_x^{\infty} ...
0
votes
1answer
29 views

solve integral equation using the theory of compact operator

Find solutions of $$u(x)-\lambda\int^{2\pi}_0\sum_{j=1}^n\frac{1}{j}cos(jy)cos(jx)u(y)dy=sin^2x$$ for all values of $\lambda$. Find the resolvent kernel for this equation. (Find the least squares ...
2
votes
0answers
20 views

Continuity of Translation and Dilation on $L^p$ spaces

Let us consider any $f \in L^p(U)$, where $U \subset \mathbb R^n$ is open, and $1 < p < \infty$. We know the translation operator $f(x) \mapsto f(x+a)$ and the dilation operator $f(x) \mapsto ...
0
votes
0answers
33 views

Integral equation with exponential

I would like to solve the following integral equation for $u(t)$, where $\theta, \gamma, \lambda, \kappa$ and $\sigma$ are parameters, but I haven't managed to obtain a solution so far. Any hints on ...
0
votes
2answers
35 views

Showing there is a projection between a normed space and a subspace

Problem: Let $E$ be a normed space. Suppose $A$ is a finite dimensional subspace of $E$. Show that there exists a continuous projection $T: E \to A.$ Proof. I can write $E=A\oplus B$, where $B$'s ...
0
votes
1answer
27 views

Funcional Analysis question [on hold]

can someone please help me solve the following question. Its very long but I would really appreciate if someone can help with any of the parts. Thanks Let $H = L^1(0,1)$ equipped with its usual ...
3
votes
1answer
162 views

A counter example

I have this two spaces $$C_{\theta}=\{u\in C(\overline{\Omega}), \sup (|x|^{\theta} |u(x)|)<\infty\}$$ with the norm $\displaystyle\|u \|_{\theta}=\sup_{\Omega}(|x|^{\theta} |u(x)|)$ and ...
0
votes
0answers
17 views

Poisson equation with nonlinear Neumann conditions

Let $\beta:\mathbb{R}\to\mathbb{R}$ be a Lipschitz function such that $0<a\leq \beta^\prime\leq b$, for some constants $a,b$. Give the weak formulation of the problem \begin{equation} ...
1
vote
1answer
13 views

Hermitian operator and numerical range

How to prove that for a complex Hilbert space $\mathcal H$ an operator $T:\mathcal H \to \mathcal H$ is hermitian if and only if it's numerical range $W(T)$ is real, where $W(T)=\{\langle Tx,x \rangle ...
9
votes
1answer
174 views

Growth of Tychonov's Counterexample for Heat Equation Uniqueness

Define a function $\varphi$ on $\mathbb{R}_{+}$ by $$\varphi(t):=\begin{cases}e^{-1/t^{2}}, & {t>0}\\ 0, & {t\leq 0}\end{cases}\tag{1}$$ It is well-known that $\varphi$ is ...
1
vote
1answer
35 views

Closed set in Baire space

I am reading a book on Banach spaces. It introduces the Baire space $\mathcal{N}=\mathbb{N}^\mathbb{N}$ as the product of infinitely many copies of $\mathbb{N}$ with the discrete topology. We have ...
2
votes
1answer
41 views

Is there an example of a non compact operator whose square is compact?

Is there an example of a non compact linear operator T from a Banach space X to itself such that T^2 is compact? Of course the converse is true, as T ^2 is compact if T is. Here T^2 means T composite ...
0
votes
3answers
37 views

Showing bounded linear operator has closed image

I'm trying to show that given a bounded linear operator $T: X \to Y$ with $X$ and $Y$ Banach such that $T$ satisfies: For ever $y \in Im(T)$ there is an $x \in X$ with $T(x) = y$ and $||x|| \le ...
0
votes
1answer
37 views

functional analysis. The spectral theorem.

How to show that if A- normal operator in H, where H-separable Hilbert space, $B_n = (Id_H + \frac{A}{n}) ^ n$ converges in the norm of $ || *||_{L (H, H)} $ to $ expA $, using the spectral theorem?
0
votes
0answers
22 views

Continuous function such that range is different from the essential range.

Let $f$ be a function $\mathbb R \to \mathbb R$. The essential range EssRan(f) of $f$ is defined as the set of all numbers $z$ such that the preimage of every open ball around $z$ under $f$ has ...
-1
votes
0answers
20 views

Hamiltonian: Scattering Spaces

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a family of projections: $$1(r)^2=1(r)=1(r)^*\quad(r\geq0)$$ Denote for shorthand: ...
2
votes
2answers
219 views

the dimension of continuous functions on a compact set is finite [duplicate]

Possible Duplicate: vector space of continuous functions on compact Hausdorff space This is a problem am trying to solve. Suppose the dimension of $C(X)$ is finite where $X$ is compact ...
7
votes
2answers
134 views

Polynomial Functional Equation.

Let $f(x)$ be a one-one, polynomial function such that $f(x)f(y)+2=f(x)+f(y)+f(xy) \ \forall \ x,y \in \mathbb R - \{0\}$, $f(1) \neq 1$, $f'(1)=3$. Find $f(x)$. I tried to find the degree of ...
2
votes
1answer
24 views

Counter-examples of direct sum of compact operators on Banach spaces is compact

Given a Banach space $X$ which can be written a direct sum of two subspaces $Y\oplus Z$ and the $u\in B(X), w\in B(Y), v\in B(Z)$ and $u=w\oplus v$. where $B(\cdot)$ denotes the space of bounded ...
-2
votes
2answers
85 views

I want some help in functional analysis [closed]

I want sone help in functional analysis : $1)$ consider the vector space $X$ of all real -valued functions which are defined on $R$ and have derivatives of all orders everywhere on $R$ define ...
0
votes
1answer
36 views

If $D$ is a dense linear subspace of $X$ then $D\to Y$ extends to $X\to Y$ uniquely

I am trying to prove the following, but I am not confident in my work. Let $D$ be a linear subspace of a normed space $X$ that is dense in $X$. Let $Y$ be a Banach space. Show that any bounded ...
0
votes
1answer
22 views

Positive operators acting on a sequence of vectors

Let $A$ be self-adjoint, unbounded operator with domain $\mathcal{D}\subset \mathcal{H}$ ($\mathcal{H}$ - Hilbert space). We assume that the spectrum of $A$ is absolutely continuous and is the set ...
3
votes
2answers
37 views

$\overline{L^2(\mathbb R)\cap L^1(\mathbb R)}^{L^2(\mathbb R)}=L^2(\mathbb R)$

While reading a proof in a book they used the following result: $$ \overline{L^2(\mathbb R)\cap L^1(\mathbb R)}^{L^2(\mathbb R)}=L^2(\mathbb R) $$ saying that it's well known !! But all I can see is ...
0
votes
0answers
27 views

Question about the spectrum of linear (unbounded) operator

I'm not much confident with functional analysis, but I found in my lecture note a statement that doesn't convince me. For a linear (possibly unbounded) operator $T$ in a Banach space the following ...
0
votes
0answers
31 views

Orthonormal List In Hilbert Space

guys say we have a orthonormal list say ${O_n}_{n\in A}$ that is an orthonormal list in a Hilbert Space $X$. Is it true that the list is complete if and only if $<a,b>= \sum_{n \in A} ...
2
votes
2answers
60 views

If $A\subset B$, then $\text{ri}\, A\subset\text{ri}\,B$?

Is it true that If $A\subset B$, then $\text{ri}\, A\subset\text{ri}\,B$? Let $u\in\text{ri}\,A$, then there is $\epsilon>0$ such that $$\mathbb B(u;\epsilon)\cap\text{aff}\,A\subset A\subset B$$ ...
-4
votes
1answer
65 views

Is Cantor set closed? [closed]

https://www.youtube.com/watch?v=dazO9UoKmyA This professor said Cantor set is closed because it's FINITE union of closed intervals at 14.00. But isn't it a wrong statement since Cantor set is ...
1
vote
0answers
20 views

Usefulness of space where metric defined but not norm.

This question is similar to the one below: Examples of metric vector spaces but not normed ? Normed but not prehilbertian? But I would like to know if there any application of the space where metric ...