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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19 views

If $u \in H^1(0,\infty)$, does $u_x(\infty) = 0$?

Let $u \in H^1(0,\infty)$. Then is it true that $u'(x) \to 0$ as $x \to \infty$? I am wondering by Green's formula/IBP in this setting; do I get something like $$\int_0^\infty u_{xx}v = ...
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6 views

Sobolev space for Mixed Dirichlet - Neumann boundary condition

Consider the subset $\Omega\subset \mathbb{R}^N$ with boundary $\partial\Omega$ sufficiently regular and let $\Gamma\subset\partial\Omega$ be a $(N-1)$- dimensional submanifold of $\partial\Omega$. ...
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0answers
7 views

Does $B_{\ell_1}$ weak-metrizable?

I know for a Banach space $X$, unit ball $B_{X}$ is weak metrizable if and only if $X^*$ is separable. My question is that Is $B_{\ell_1}$ weak-metrizable?
3
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20 views

Isolated Eigenvalue

What does it mean that an eigenvalue is "isolated"? My intuitive understanding says it is when one can find an open ball around it such that there is no other eigenvalue in that open ball. However, I ...
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0answers
13 views

Intuition of weak star convergence.

Given $\Omega=(0,1)$, consider the following sequence $$ v_j(x)\colon=\begin{cases} \;a &\text{if }jx-\lfloor jx \rfloor\le\theta\\ \;b &\text{otherwise} \end{cases} $$ where ...
2
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40 views

Functional Maximization

So how do we solve a problem like this: Find the function $s(x)$ such that $s(x)$ maximizes $$\int_0^{s^{-1}(k)} s(x) dx $$ where $x\in[0,10]$, $s(x)\in[0,1]$, and $k\in[0,1]$ ($k$ is a constant). ...
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1answer
24 views

Eigenfunctions of $-\Delta_{S^n}$

I read somewhere that the eigenvalues of the Laplacian $-\Delta_{S^n}$ on the sphere $S^n$ consist of $k^2 + (n - 1)k$, with the corresponding eigenspace $V_k$ consisting of homogeneous harmonic ...
2
votes
1answer
14 views

Existence of Certain Locally Integrable Function Defining a Tempered Distribution

We say that a distribution $T$ is tempered if for every sequence $\left\{\phi_{n}\right\}$ in $C_{c}^{\infty}(\mathbb{R}^{n})$ tending to $0$ in the topology of the Schwartz space ...
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2answers
20 views

Show that properties of norm are satisfied

Show that \begin{align} & \|y\|_M= \max_{a \leq x \leq b} |y(x)| \tag 1 \\[8pt] & \|y\|_1=\int_a^b |y(x)|\, dx \tag 2 \end{align} satify the properties of a norm in $C[a,b]$. That's what I ...
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0answers
17 views

How do I compute this metric projection?

I saw a result that says: Given a nonzero vector $a$ and the convex set $K:=\{y\in H: \langle a,y\rangle =\alpha, \alpha \in \mathbb{R}\}$ a hyperplane, then $$P_Kx=x-\frac{\langle ...
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1answer
27 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: http://www.encyclopediaofmath.org/index.php/Ill-posed_problems 1) Can there be a class of ...
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2answers
58 views

Is $[-1,1]$ complete under the Euclidean metric? [on hold]

Is it true that the interval $[-1,1]$ is complete under the Euclidean metric?
2
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1answer
24 views

Normed space where unit ball's weak and norm topology coincide?

Let $X$ be a normed space. Are there any infinite-dimensional examples such that the $(B_X, w\restriction_B) = (B_X, \|\cdot \|\restriction_B)$. In finite dimensions this obviously holds always true. ...
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0answers
8 views

under what conditions, a function on $CB(X)$, space of closed and bounded sets, [on hold]

under what conditions, a function on $CB(X)$, space of closed and bounded sets, will be continuous. where function $F$ defined on $CB(X)$ as F(A)={\bigcup F(x):x∈A}
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6 views

Deriving information about asymptotics from finitness of a limit

Let $f_1,f_2:\mathbb{R}\setminus\{0\}\to \mathbb{R^+}$ be two $C^1$ functions and $\alpha:\mathbb{R}\setminus \{0\}\to \mathbb{R}$ be a function from a Zygmund class (in particular it is Holder for ...
0
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1answer
16 views

Exponential decay estimate involving $C_{0}$-semigroup and the principle eigenvalue

Consider the heat equation defined as $$\begin{cases} \partial_{t}u(x,t)=\Delta_{x}u(x,t) &\mbox{ } t>0\mbox{, } u\in U \\ u(x,t)=0 &\mbox{ } t\ge 0\mbox{, } x\in\partial U \\ u(x,0)=g(x) ...
3
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3answers
38 views

Show $\langle u,v \rangle = -\frac{1}{2}$ when $u+v+w=0$ and $\|u\|=\|v\|=\|w\|=1$?

Show $\langle u,v \rangle = -\frac{1}{2}$ when $u+v+w=0$ and $\|u\|=\|v\|=\|w\|=1$? My thinking is: $\langle u+v+w,v \rangle =0 \iff \langle u,v \rangle + \langle w,v \rangle = -1$ How do i ...
2
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0answers
18 views

Multiplication operators are sectorial

Consider the multiplication operator $M_a$ on $L^p(M)$, where $M$ is a Riemannian manifold, and $a$ is a non-negative function. An operator $A$ is said to be sectorial if there exists $\theta \in (0, ...
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15 views

function defined as integral of borel function

I know that $f \in B_b(E)$, where $B_b(E)$ is the set of Borel bounded function on an euclidean space E. I have to show that: \begin{equation} x \to \int_{0}^{+\infty} e^{-at} P_tf(x) dt ...
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20 views

For $z=(c_n)\in l^3$ and $N(z)=\left(\sum_{n=1}^{\infty} \left|\frac{c_n}{n}\right|^3\right)^{\frac{1}{3}}$. Show $N(z_1+z_2)\leq N(z_1)+N(z_2)$?

For $z=(c_n)\in l^3$ and $N(z)=\left(\sum_{n=1}^{\infty} \left|\frac{c_n}{n}\right|^3\right)^{\frac{1}{3}}$. Show $N(z_1+z_2)\leq N(z_1)+N(z_2)$? Obviously $$N(z_1+z_2)=\left(\sum_{n=1}^{\infty} ...
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19 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 $\langle Px,y \rangle=\langle x,Py \rangle \forall x,y\in H$. ...
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0answers
18 views

Is there any apps for downloading study materials of our mathematics papers for exam preparation [on hold]

I need apps details for downloading study materials for exam preparation
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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 ...
3
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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 ...
0
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1answer
31 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 $A_k=\frac{|x_k|}{\|x\|}$ ...
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0answers
20 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 ...
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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 ...
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0answers
15 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
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18 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
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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 ...
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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 ...
2
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1answer
27 views

Showing that an operator generates a contraction semigroup

Let $A$ be the infinitesimal generator of a contraction semigroup $(T(t))_{t\ge 0}$ on the Hilbert space $X$, and $D\in\mathcal{L}(X)$. I want to show that the operator $A+D-2\|D\|I$ with domain ...
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0answers
19 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 ...
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2answers
27 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 ...
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0answers
18 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
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0answers
32 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 ...
2
votes
1answer
45 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 $$d(x;\Omega)=\inf\{||x-\omega||\mid\omega\in\Omega\}$$ ...
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0answers
15 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} ...
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0answers
22 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 ...
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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} ...
2
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2answers
56 views

Definition of convergence in the product and in the box topology

I am really struggling with the differences between convergence in the product topology and convergence in the box topology. More specifically, I have some doubts concerning the definitions of those ...
0
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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 ...
0
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0answers
27 views

Functional Analysis, a question that needs clarification.

Find the norm of the linear operator $A:C[-1,1]\to L^p[-1,1]; p\geq1$ that is defined as: $$A(x(t))=\int_{-1}^{1}{{x(s)\over (t-s)^{1 \over 3}}}ds$$ Can someone provide an answer with a little more ...
0
votes
1answer
23 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
1answer
22 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 ...
1
vote
1answer
18 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 ...
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 ...
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votes
2answers
32 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 ...
2
votes
1answer
40 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
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0answers
17 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: ...