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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8 views

Injective linear endomorphism of hilbert space is bijective?

Is it true that an injective continuous endomorphism of a hilbert space is bijective? If not, are there conditions that imply this? I know this would follow from the rank nullity theorem in finite ...
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0answers
23 views

limit problem-equation

H, I have this problem $$c^2 U''(x)=F(x),\quad U(0)=A,\quad U(\ell)=B$$ $F$ is done, and $0 < x < \ell$ I read that we must found that $$U(x) = A + (B-A)\frac{x}{\ell} + \dfrac{x}{\ell} ...
2
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1answer
32 views

Borel regular measure: Approximate any measureable set by compact sets

Let $(K,\mathcal{F},\mu)$ be a measure space. Let $K$ be a compact Hausdorff space and $\mu$ be a regular finite measure. We said that it is regular if $\mu(A) = \inf\{\mu(B): B \text{ open }, ...
3
votes
2answers
44 views

Non-compactness of the resolvent

Consider a complete non-compact Riemannian manifold $M$ and the resolvent of the Laplacian $(-\Delta + \lambda I)^{-1}$. It is known that the resolvent is in general not a compact operator. I am ...
1
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1answer
24 views

A Banach space in between $L^{1}$ and $L^{2}$, does it make sense?

Let $L^{p} (A, B)$ be a collection of functions $f:A \mapsto B$ satisfying $$(\|f\|_{p})^{p} := \int_{A} |f(x)|^{p} dx <\infty.$$ Now we consider functions $f:[0,1]^{2} \mapsto [0,1]$. We say ...
1
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0answers
15 views

Continuously bounding a matrix valued function by another matrix valued function

Notation: The symbol $\mathbb{S}^{n \times n}$ denotes the set of symmetric matrices of order $n \times n$ and $A \succ 0$ denotes that $A$ is a positve definite matrix. Let $M:S \rightarrow ...
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43 views
+50

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: $$A(x(t))=\int_{-1}^{1}{{x(s)\over (s-t)^{1 \over 3}}}ds$$ Can someone provide an answer with a little more ...
1
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0answers
21 views

$L^2$-Sobolev space

I am looking at the proof of the following lemma and I don't understand the conclusion. Lemma: For $k\geq 0$ integer, let $f=\sum_{n\in \mathbb{Z^d}}{c_n\chi_n}\in L^2(\mathbb{T}^d)$. Then $$f\in ...
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4answers
66 views

Is an non-invertable matrix an linear operator?

I am under the impression that any matrix can be called a linear operator, even if the matrix does not have an inverse. Is it true? There are many properties a linear operator enjoys; do all matrices ...
2
votes
2answers
18 views

Convex cone of nonnegative functions in L2 has empty interior

Convex cone $S:=\{f\in L^2(\mathbb{R},\mu):f\geq 0\}$ has empty interior in $L^2(\mathbb{R},\mu)$ when $\mu$ is Lebesgue measure. I wanted to prove it but i have major holes in my knowledge of ...
2
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1answer
315 views

Double orthogonal complement of any closed subspace is it self

Let $H$ be a pre-Hilbert space such that any closed sub space $M \subset H$ has the property $M^{\bot \bot}=M$. Prove that $H$ is a Hilbert space (ie, prove that $H$ is complete) My attempt: As ...
0
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1answer
24 views

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

I know that for a Banach space $X$, the unit ball $B_{X}$ is weak metrizable if and only if $X^*$ is separable. My question is that Is $B_{\ell_1}$ weak-metrizable?
4
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1answer
40 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 ...
3
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1answer
31 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 ...
3
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0answers
22 views

When can we exchange the trace and an integral/limit/derivative?

For a trace class operator $A$ (acting on a Hilbert space), that is parameterised by a real variable $x$, what are the conditions for the following to hold? $$ \mathrm{tr} \int_a^b A(x) \, dx = ...
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1answer
13 views

symetric closed operator and extension [on hold]

i have this question let A a symetric closed operator let pose that A have a self adjoint extension is possible that A has an extension such that closure A can't have a self adjoint extension
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0answers
15 views

Two-dimensional subespace suplementary of another one

Let E be a real normed space. All subespace S of codimension 1 (hyperplane) in E is either, closed or dense. What do say about a similar property when S is of codimension 2?
1
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1answer
18 views

Completely positive maps

Let $B$ be a commutative C$^*$-algebra and let $M_n$ denote the algebra of $n\times n$ complex matrices. Let $f$ be a state on the tensor product of $B$ and $M_n$, $B\otimes M_n$. How can I show that ...
1
vote
2answers
25 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 ...
1
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1answer
39 views

Lemma 3.5-3 in Erwine Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Is the set of non-zere Fourier co-efficients uncountable too?

Let $X$ be an inner product space, let $x \in X$ be non-zero, and let $M$ be an uncountable orthonormal subset of $X$. Then what can we say about the cardinality of the following set? $$ \{ \ v \in M ...
3
votes
2answers
42 views

An infinite dimensional normed linear space is the union of two disjoint convex sets

Let $X$ be an infinite dimensional normed linear space. I want to show that there exist two disjoint convex sets $C_1$ and $C_2$ such that $X=C_1\cup C_2$ and both $C_1$ and $C_2$ are dense in $X$. I ...
1
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1answer
36 views

Normal Operators: Transform

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(T)\to\mathcal{H}:\quad N^*N=NN^*$$ Construct the operator: $$W:=(1+N^*N)^{-1}:\quad Z:=N\sqrt{W}$$ Then it is ...
0
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1answer
39 views

Show that the functional is continuous everywhere in $V$

Let $J: V \to \mathbb{R}$ be a linear functional and $V$ a linear space with norm. Show that if $J$ is continuous on $0 \in V$ then $J$ is continuous everywhere in $V$. That's what I have tried: ...
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0answers
28 views

a question on Frechet derivative

Suppose the derivative of a functional is given by $\int_{\Omega}(\vec{v}.\nabla u)|\nabla u|^{p-2} \phi$ for $\phi\in W_0^{1,p}(\Omega)$, then what is the functional?.
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1answer
330 views

Equality in the Cauchy-Bunyakowsky-Schwarz inequality for a semi-inner product

I am stuck with an exercise that I found in a textbook by Conway. First, I would like to clarify what is meant by a semi-inner product. Definition. Suppose that $\mathscr X$ is a vector space over ...
0
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0answers
12 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 $\partial\Omega$. ...
0
votes
1answer
30 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 = ...
7
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2answers
96 views
+50

Hölder continuous functions are of 1st category in $C[0,1]$

I'm trying to show that the Hölder continuous functions in $[0,1]$ are a set of first category in $C[0,1]$. Does it suffice to show that they are not an open subset of $C[0,1]$? Let ...
0
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1answer
32 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 ...
2
votes
1answer
22 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 ...
1
vote
1answer
29 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
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1answer
72 views

$\oint_{C}(A-\lambda I)^{-1}\,d\lambda=0$ implies interior of $C$ is in the resolvent.

Suppose that $A$ is a bounded linear operator on a complex Banach space $X$ with resolvent set $\rho(A)$. If $C$ is a simple closed smooth curve in $\rho(A)$ such that $$ ...
2
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0answers
52 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). ...
2
votes
2answers
80 views

Prove that the set $X+Y$ is closed

Consider two sets: $$X=\bigl\{(x,y)\in \mathbb R^{2}:xy=1\bigr\}$$ $$Y=\bigl\{(x,y)\in \mathbb R^{2}:|x|\le 1,|y|\le 1\bigr\}.$$ We find that , both the sets are closed. But, is the set $X+Y$ ...
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0answers
20 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 ...
0
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2answers
60 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
votes
1answer
27 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. ...
2
votes
0answers
23 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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votes
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}
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 ...
2
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0answers
33 views
+50

Why has the Stein operator for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$?

My Question: Why has the Stein operator $\mathcal A$ for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$? How can one deduce this form of the operator? Reason for my question: I ...
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0answers
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 ...
2
votes
1answer
29 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 ...
0
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1answer
18 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
votes
3answers
39 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 ...
0
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0answers
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 ...
0
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0answers
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} ...
2
votes
0answers
75 views

Besov–Zygmund spaces and the Inverse Function Theorem, is the Inverse Zygmund?

Preliminary Definitions Let $\Omega \subset \mathbb{R}^n$ be open. We define the Zygmund spaces $C^r_{*}(\Omega)$ with $r>0$, $r \in \mathbb{R}$ in the following way: (all the functions are ...
30
votes
2answers
547 views

Prove or disprove a claim related to $L^p$ space

The following question is just a toy model: Let $f:[0,1] \rightarrow \mathbb{R}$ be Lebesgue integrable, and suppose that for any $0\le a<b \le1$, $$\int_a^b |f(x)|dx \le \sqrt{b-a}$$ then prove ...
16
votes
2answers
177 views

Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponent allows for convenient ...