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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58 views

Translation invariance and finite dimension imply smoothness

Let $X$ be linear subspace of $C(\mathbb R)$, the set of continuous functions on $\mathbb R$, which is closed under translations, i.e., if $f\in X$ and $h\in\mathbb R$, then $\tau_h f\in X$, where ...
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2answers
91 views

Banach space valued random variable

Let $X$ be a Banach space valued random variable. Is there a characteristic function of $X$ in this case? How it is defined if there is one?Are there any applications of this function in this high ...
1
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1answer
107 views

Gagliardo Nirenberg Sobolev inequality for n >= 2

I have a quick question regarding the Gagliardo-Nirenberg-Sobolev inequality. It states that: Assume $1 \leq p < n$. There exists a constant $C$, depending only on $p$ and $n$, such that ...
8
votes
2answers
766 views

$L^1$ and $L^{\infty}$ are not reflexive

I want some proof for the following statement : $L^1$ and $L^{\infty}$ are not reflexive. Can anyone help me, please? or reference me?
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0answers
58 views

$L^{\infty}$ is p-concave

I want to show that the $L^{\infty}$ on a Banach lattice $X$ is $p$-concave with $M_{(p)}(L^{\infty})=1$. Where $L^\infty=L^\infty(X,\mathcal{M},\mu)$. Recall that a Banach lattice $X$ is said ...
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1answer
54 views

About absolutely summing operators

The concept of absolute summing operator between two Banach spaces is already well-known. I just would like to know if one can extend the definition between general spaces, say locally convex spaces. ...
7
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2answers
81 views

Detecting compactness from the ring of smooth functions

Given a smooth manifold $M$, is there some ring-theoretic property (preferably not mentioning $M$) such that $C^{\infty}(M)$ has this property if and only if $M$ is compact?
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4answers
65 views

If $\textrm{dim}(V)<\infty\Rightarrow \textrm{dim}(U)<\infty$?

Consider $U$ and $V$ two vector spaces over a given field. $T:U\longrightarrow V$ and $S:V\longrightarrow U$ two linear operators such that $S\circ T=Id_U$. If $\textrm{dim}(V)<\infty$ how can I ...
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0answers
40 views

$\gamma-$radonifying operators.

I am reading about $\gamma$-radonifying operators, and came across 2 similar definitions. And I hoped to understand why they are equivalent. Let $H$ be a seperable real Hilbertspace, $E$ banach ...
0
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1answer
61 views

Using Cauchy Schwarz for functions in Sobolev Space

Hi I just want to confirm something simple and check that the following is allowed: The Cauchy-Schwarz inequality states if $A = ((a_{ij}))$ is a symmetric, non-negative $n \times n$ matrix then ...
0
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3answers
67 views

Positive unbounded operator with zero not as an eigenalue

I am currently doing Quantum Mechanics and I am supposed to show that zero is an eigenvalue of a positive operator. I have no knowledge of Functional Analysis at that kind of level, so I was wondering ...
3
votes
1answer
76 views

When is it possible to construct ladder operators for a given Hamiltonian?

It is pretty cool (in my opinion) that one can solve Schrödinger's equation for the harmonic oscillator by using ladder operators, rather than just integrating it. In particular, it is possible to ...
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3answers
66 views

Two parts I cannot understanding on simple proof about Hilbert Space

I am currently studying Hilbert Space in Real analysis, and I have a part not understandable. This is a theorem for Hilbert Space $H$. $Theorem$ : If $L$ is a bounded linear functional on $H$, then ...
1
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1answer
73 views

completely continuous implies compact

I'm searching for a proof of the fact that if: $T$ is a bounded operator in a reflexive Banach space that maps weakly convergent sequences onto convergent sequences then $T$ is compact. If we let ...
3
votes
2answers
62 views

Is this operator compact and how do I prove it? [duplicate]

I have a very big problem with the following question: Is the operator $T$ defined by $(Tx)t=tx(t)$, $(0<t<1)$ compact in $L_2(0,1)$? My guess is no and I've tried 3 different approaches to ...
2
votes
1answer
62 views

Closed range in Hilbert Space

If $H$ is a Hilbert Space. Let $A: H \rightarrow H$ be a one-to-one bounded operator with the additional property that $\beta||u|| \leq ||Au||$. How would you show that $R(A)$ (the range of A) is ...
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1answer
59 views

A question about range projection in von Neumann algebra.

I am reading a book about C*-algebra. And I meet with a problem. Recall the range projection of an operator $a\in B(H)$ is the projection on the closure of $\{a(\eta):\eta\in H\}$(Here, $H$ is a ...
4
votes
1answer
158 views

Weak solution $u(x,t)$ of heat equation converges as $t \in \infty$

Where can I find a proof that the weak solution $u \in L^2(0,T;H^1) \cap H^1(0,T;H^{-1})$ of the heat equation $$u_t -\Delta u = f$$ converges as $t \to \infty$ to the solution of the elliptic PDE ...
2
votes
2answers
121 views

Problem with spectral theorem and spectral measure.

There is a passage in a book that is not very clear to me: A is a C*Algebra and $a$ is selfadjoint. Then "Indeed identifying A with an algebra of operators on a Hilbert space $\mathcal{H}$, by the ...
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0answers
43 views

How do I prove this secondary functional derivative?

This problem is from a book; I can't solve it. Please help. Let $$J=||R-SG^T\bigodot ...
2
votes
1answer
79 views

Haar measure, convolution and involutions

I have some problems to follow the proof of the anti commutativity property of the convolution and involution operations defined using a Haar measure as presented in Pedersen's book "analysis Now", ...
4
votes
1answer
330 views

Counterexample for the Open Mapping Theorem

I would like to ask a counterexample for the open mapping theorem: Find a discontinuous linear mapping $T:X \to Y$ such that $T(X)=Y$ and $X,\;Y$ are Banach but $T$ is not open. Could you help me ...
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1answer
63 views

prove the existence of a measure $\mu$

Suppose $X$ and $Y$ are compact metric spaces and $F : X \rightarrow Y$ is a continuous map from $X$ onto $Y$. If $\nu$ is a finite measure on the Borel sets of $Y$, prove that there exists a measure ...
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1answer
46 views

Question on representation of a Banach algebra

Let $A$ be a Banach algebra and $\pi$ a continuous irreducible representation of $A$ on a Banach space $X$. Suppose $\pi(a)\xi\neq0$ for some $a\in A$ and some $\xi\in X$. Let $\eta\in X$. The ...
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1answer
43 views

Difference between $H_0^2$ and $H^2 \cap H_0^1$

What's the difference between Hilbert space $H_0^2$ and $H_0^1 \cap H^2$?
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1answer
20 views

Nonlinear isometric embedding of $ \mathbb{R} $ to $ (\mathbb{R}^2,\|.\|_\infty) $

Prove that there is an isometric embedding of $ \mathbb{R} $ to $ (\mathbb{R}^2,\|.\|_\infty) $ that is not linear. Thanks.
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0answers
31 views

Derivative of infimum in variational problem

Let $\mathcal{E}(\phi,\alpha), \phi\in \mathcal{D}$ be a functional on some domain $\mathcal{D}$ that depends on a parameter $\alpha$. In the expression $$\frac{\partial}{\partial \alpha} \inf_{\phi ...
0
votes
1answer
22 views

Does $v\in L^p$ imply the derivative or integral of $v$ also in $L^p$?

And furthermore, will the composition $(voF)$ be also in $L^p$ if $F$ is bijective and $v\in L^p$.
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0answers
23 views

Exchange limit and infimum in variational problem

Let $\{\mathcal{E}_n\}_{n\in \mathbb{N}}$ be a sequence of functionals over the same domain $\mathcal{D}$. What are sufficient conditions on the sequence and possibly the domain such that ...
1
vote
1answer
44 views

Redundance in $l^p$ space.

I am covering the following problem: Let $A \subset l^p$, with $p \in [1,\infty)$, then is equivalent: i)A is relatively compact ii)A is bounded and we have $$\lim_{n \rightarrow \infty} \sup_{x ...
2
votes
1answer
57 views

Norm of the dual of the Tensor product of Hilbert spaces

Let $V$ and $W$ be Hilbert spaces, we can define inner product and induced norm on Tensor product of these spaces as: Let $v_1,v_2 \in V$,and $w_1,w_2 \in W$. then $(v_1 \otimes w_1, v_2 \otimes ...
0
votes
1answer
48 views

Prove differentiability of functional.

In $C[0;1]$ space let's consider following functional: $$\phi(f) = \int_{0}^{1}(1+f(t))^{3}dt.$$ Prove differentiability of $\phi$ and find $\mathrm{D}\phi(f)$ for: $f(t)=0$, $f(t)=t$, $f(t)=\cos t$. ...
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vote
1answer
282 views

Polar decomposition normal operator

Let $T \in \mathcal{B}(\mathcal{H})$ be normal. I have to show that there exists a unitary operator $U \in \mathcal{B}(\mathcal{H})$ such that $T^*=UT$ and give necessary and sufficient conditions on ...
2
votes
1answer
42 views

spectral mapping type norm identity for self adjoint operator

I am currently trying to understand the spectral theorem as given in "Functional Analysis" (Vol.1) by Reed and Simon. Leading to its proof is a preliminary Lemma where I got stuck. It says Let $P(x) ...
2
votes
2answers
45 views

Compact operator and limit

I was wondering about something related to compact operators. If we have a compact operator $T:X \mapsto Y$ and a bounded sequence $(x_n)n$, then we know that there is a convergent subsequence ...
0
votes
1answer
40 views

What is a countably infinite-dimensional coordinate space called and where can I read more about it?

This is just a question about terminology. I'm interested in spaces of vectors with countably many components $(x_{1},x_{2},x_{3},\ldots)$ where each $x_{k} \in \mathbb{R}$. What are these spaces ...
1
vote
1answer
78 views

Prove this is a bounded linear operator and find its operator norm?

I have a map $$A:(C[0,1], || \centerdot ||_\infty) \rightarrow \mathbb R, Ax = x(0) \forall x \in C[0,1]$$ and need to prove it's a bounded linear operator, and find its operator norm. I've tried ...
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vote
1answer
51 views

Different definitions of Banach lattices?

Do I get it correctly, that the are different definitions of "Banach lattices" available in the literature? To be precise, some authors (like Schaefer) include the order continuity of the norm, while ...
2
votes
1answer
62 views

Are these operators and the fourier transform compact?

I do not want a proof but rather an explanation. I just read that $T_k:L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})$ such that $(T_kf)(s) = \int_{\mathbb{R}} k(s,t)f(t) dt $ is compact. (in this ...
0
votes
2answers
99 views

Polar Decomposition and Compact operator

Let $H$ be an infinite dimensional separable Hilbert space and $\{e_n\}$ be a countable orthonormal basis for $H$.For a bounded sequence $\{a_n\}$ define $T(e_n)=a_ne_{n+1}$ and extend linearly to ...
1
vote
0answers
91 views

Closed graph proof and existence of continuous function

I'm working on an exercise in which you need to prove the existence of a certain function, but I'm not quite sure how to do this. A previous assignment had a similar proof that I couldn't figure out ...
6
votes
1answer
157 views

Definition of the Hamiltonian via Legendre transform.

In my book of classical mechanics (Mathematical methods of classical mechanics by V.I. Arnold), the Hamiltonian is introduced in this way (my translation): Let us consider the system of equations ...
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vote
0answers
49 views

How to show that there do not exist bounded operators $A$ and $B$ such that $AB-BA=I$ [duplicate]

Suppose $X$ is a Banach space, $A$ and $B$ are bounded operators on $X$. Then how to show that it's impossible to have $$AB-BA=I$$ If $A$ and $B$ are matrix, then we just take trace of both side to ...
0
votes
1answer
34 views

Holder continuity and Hilber space

Let $\Omega\subset \Re^n$ be an open set and let $u \in H^1_{loc}(\Omega)$ be a weak solution of $\Delta u=f $ in $\Omega$, with $f \in C^{0,\alpha}(\Omega)$. Prove that $u \in C^{2,\alpha}(K)$ for ...
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votes
3answers
69 views

Compute the norm of matrix

Let $M$ be $n\times n$ matrix, consisting entirely of 1's. Show, that $\|M\|_{op}=\sup_{x\in C^n}|Mx|=n$.
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vote
1answer
48 views

supremum norm and convergence.

Suppose $f:[0,\infty) \rightarrow \mathbb{R}$ is continuous. Suppose that for some $\epsilon$ > 0, $\max_{t \in [0,n]} |f(t)|$ < $\epsilon$ for all $n \in \mathbb{N}$. Is it then true that ...
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2answers
57 views

Positive Operator on Hilbert Space

I am stuck with this problem for a long time. If $T\geq0$ then show that $T^2\leq$||T||T. Any help will be appreciated.
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0answers
33 views

A domain in which the dirichlet laplacian has eigen values of all orders

I am trying to come up with an example to the following. Construct a domain $\Omega$ in all dimensions $n \in \mathbb{N}$, such that the spectrum of the Dirichlet Laplacian on such a domain (i.e., ...
1
vote
1answer
19 views

$\mathcal{l}^1$ space and proving a graph is closed

Let $X= \{\xi =(x_n)_{n \in \mathbb{N}} \in \mathcal{l}^1 : \sum_{n=1}^{\infty} n |x_n| < \infty \}$ And define the map $T: X \to \mathcal{l}^1$ by $(T \xi)_n := n x_n$ I'm trying to proof that ...
4
votes
0answers
73 views

Regularity theorem for Laplacian

Let $\Omega \subset \mathbb R^d$ be a bounded domain, $d>2$. Let $f \in C^\infty(\Omega)$. If $u \in L^2$ is a distributional solution of $\Delta u = f$ in $\Omega$ then $u \in C^\infty(\Omega)$ ...