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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2k views

Continuous projections in $\ell_1$ with norm $>1$

I was trying to find papers and articles about non-contractive continuous projections in $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the case ...
10
votes
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91 views

When does a real function together with its derivatives form a basis?

Consider real functions defined on an interval, let's say $[-\pi,\pi]$. Let $s(x)$ be an analytic, periodic function such that all of its Fourier components are non-zero. We build a set of functions ...
9
votes
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372 views

Compact set of probability measures

I think I can solve the following exercise if X is assumed to be separable, otherwise I can't. Let $X$ be a (Hausdorff) locally compact space, $\pi\colon X \to Y$ a continuous map into a ...
9
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744 views

Eigenvalues and eigenfunctions for the Fredholm integral operator $K(g) = \int_0^1 e^{x t} g(t) dt$.

I would like to compute the eigenvalues and eigenfunctions for the Fredholm integral operator $$K(g) = \int_0^1 e^{xt} g(t) dt.$$ The sources I've checked* seem to say that the process is fairly ...
8
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0answers
87 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 exponenent allows for convenient ...
8
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160 views

Gradient Estimate - Question about Inequality vs. Equality sign in one part

$\DeclareMathOperator{\diam}{diam}\newcommand{\norm}[1]{\lVert#1\rVert}\newcommand{\abs}[1]{\lvert#1\rvert}$For $u \in C^{1}(\overline{\Omega})$, for $\Omega\subset \subset \mathbb{R^{n}}$ a bounded ...
8
votes
0answers
396 views

Finding a proper solution of a given functional

It's my first post here, but I worked very hard to find solution and I failed. Hereinafter, I skip physical background and directly proceed to my mathematical problem. No matter how, you know the ...
6
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70 views

Showing L infinity norm bounded by L2 norm on a manifold

I have the following problem that I'm working on: Suppose $(M, g_{ij})$ is a compact Riemannian manifold. Assume $u$ is a smooth, nonnegative function which satisfies the differential inequality ...
6
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95 views

Two definitions of $H^1(\partial\Omega)$, one using charts and one use tangential gradients

Let $\Omega$ be a bounded Lipschitz domain with boundary $\partial\Omega$. There are two ways to define a space $H^1(\partial\Omega)$: By using charts, we can define $H^1(\partial\Omega)$ to ...
6
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126 views

Heat Kernel Property

Let $\phi$ be the Heat Kernel in $\mathbb{R}^n$. and let $u$ satisfy Heat equation. Show that: $$\frac{d}{dt}\int\limits_{\mathbb{R}^n}\phi |Du|^2=-2\int\limits_{\mathbb{R}^n}(\Delta u)^2$$ What I ...
6
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0answers
86 views

References on the Nash-Moser Implicit Function Theorem

To learn, the Nash-Moser implicit function theorem, I tried with Hamilton (1982) The Inverse Function Theorem of Nash and Moser. But, the article is very encyclopedic. I have a background in ...
6
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130 views

Explicit form of the homeomorphism between $C[0,1]$ and $C[0,1]\setminus 0$

How to construct the homeomorphism between $C[0,1]$ and $C[0,1]\setminus\{\theta\}$ in the explicit form? Here, as usual, $C[0,1]$ is the Banach space of continuous functions $f:[0,1]\to\mathbb{R}$ ...
6
votes
0answers
131 views

Existence of a map in a Hilbert space

Let $H$ be an infinite-dimensional Hilbert space, $B$ be its unit ball: $B=\{x\in H: \, \|x\|\leq 1\}$. Does there exist a continuous map $f:H\to H$ such that $f(f(x))=x$ $\forall x\in H$, $f$ has no ...
6
votes
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383 views

On the weak and strong convergence of an iterative sequence

I have some difficulties in the following problem. I would like to thank for all kind help and construction. Let $H$ be an infinite dimensional real Hilbert space and $F: H\rightarrow H$ be a ...
6
votes
0answers
1k views

Are vague convergence and weak convergence of measures both weak* convergence?

For quite a long time, I have been confused about the definitions of weak convergence and vague convergence of measures among other modes of convergence that root from functional analysis, mainly due ...
6
votes
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139 views

Isomorphism between spaces of sections.

Let $M$ be a compact manifold and let $E_i \xrightarrow{\Large \pi_i} M$, $i = 1, 2$, be two (real or complex) vector bundles of the same rank $k$ over $M$. Assume we have metrics $g_1, g_2$ on $E_1$, ...
6
votes
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269 views

Hilbert transform and Hilbert matrix

The Hilbert matrix is \begin{bmatrix} 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \dots \\[4pt] \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & & \ddots \\[4pt] ...
6
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145 views

Closure in the Space of Probability Measures with the Prohorov metric

I have seen this result stated countless times: assume the metric space $(\theta,d)$ is separable; then $(\theta,d)$ is complete if and only if the space $(\mathcal{P}(\Theta),\rho)$ (the space of ...
6
votes
0answers
122 views

Decomposing $\mathcal{B}(H)$

Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
6
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1k views

Prerequisites for functional analysis

I'm having a few months of free time, and I decided to do a self-study on functional analysis (in-depth) in the meantime. I'm aware that functional analysis requires a good deal of foundation from ...
6
votes
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150 views

Invertibility of Toeplitz operator in $\ell_1$

Suppose we have a Toeplitz operator $$ T(a) = \begin{bmatrix} a_{0} & a_{-1} & a_{-2} & \ldots & \ldots &a_{-n+1} &\dots \\\\ a_{1} & a_0 & a_{-1} & \ddots & ...
6
votes
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144 views

Linear isomorphisms with dense graph

Is it true that for each infinite dimensional Banach space $X$ there exists a linear bijection $f: X \rightarrow X$ with a dense graph? A graph of $f$ it is the set $\Gamma(f):=\{(x, f(x)): x \in X ...
6
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292 views

Fixed point: linear operators

I ask my question in two parts: though the topic is similar, I would like to distinguish linear and general cases since methods may be too different while my questions are broad. Consider a space $X$ ...
6
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0answers
187 views

Two “different” adjoints of exterior derivative on manifolds with boundary in the $L^2$-setting

The follow problem appears in the setting of $L^2$-differential forms on manifolds with boundary. An abstracted operator theoretic problem is given below. Suppose $M$ is a smooth Riemannian manifold ...
5
votes
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99 views

Sobolev spaces and using monotone convergence theorem (don't understand a paper)

I'm reading this paper. In it there the following argument (see page 240). Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. ...
5
votes
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99 views

Connections and dependences between topological and algebraic basis in topological vector space

On my last functional analysis exam, one of the tasks was to show that if normed vector space $X$ have countable Hamel basis, then $X$ is separable space (over field $\mathbb{R}$). I am not sure if ...
5
votes
0answers
68 views

Inner product on $C(\mathbb R)$

With Axiom of choice it is possible to construct an inner product on $C(\mathbb R)$. Of course, the space would be not complete under the norm induced by the inner product. My question is, is it ...
5
votes
0answers
148 views

Subsequences of a basic sequence

Suppose ($x_n$) is a basic sequence in a Banach space $X$, and $Y$ is a closed, infinite co-dimensional subspace of the closed span of $(x_n)$. Can we always find a subsequence ($y_n$) of ($x_n$) such ...
5
votes
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67 views

Does this sequence of operators converge in norm or strongly?

Let $H$ be a Hilbert space and $\mathcal{L}(H)$ the set of all bounded linear operators $L:H\to H$, equiped with the usual norm $\|\cdot\|_{\mathcal{L}}$. Let $T:D(T)\subset H\to H$ be a ...
5
votes
0answers
87 views

Gradient on Sobolev spaces

Suppose I have a function $\Phi \in L_p(\Omega)$ and $\nabla\Phi \in W^{-\alpha}_p(\Omega)$, $1<p<\infty$ and where $\Omega\subset\mathbb{R}^n$ is a bounded domain. Can I conclude that $\Phi\in ...
5
votes
0answers
53 views

Radon-Nikodým (write the density as a limit)

Let $\mu$ be a probability measure and $\nu$ a $\sigma$-finite measure on $(\mathbb{R},\mathcal{B})$ with $\nu\ll\mu$. Show that it is $\mu$-a.s. $$ \lim_{h\to 0}\frac{\nu [x-h,x+h)}{\mu ...
5
votes
0answers
104 views

renorm a Banach space to make an operator have spectral radius equal to norm

Let $X$ be an infinite-dimensional complex Banach space equipped with the norm $\lVert\cdot\rVert$, and let $T\in\mathcal{L}(X)$ a bounded linear operator on $X$. Let $r(T)$ denote the spectral ...
5
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33 views

Closed Monoidal Structures On The Category Of Complete Topological Vector Spaces

Context: The category of Banach spaces, with the projective tensor product is a closed monoidal category. Question 1: Is there a tensor product on the category of complete topological vector spaces, ...
5
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0answers
135 views

Dyadic approach to Marcinkiewicz interpolation for Lorentz Spaces

In Exercise 21 , in a note for professor Terrence Tao on his own blog http://terrytao.wordpress.com/2009/03/30/245c-notes-1-interpolation-of-lp-spaces/ Exercise 21 Suppose we are in the ...
5
votes
0answers
161 views

If a Radon measure is a tempered distribution, does it integrate all Schwartz functions?

The question might at first sight sound like the answer is trivially "yes", so let me clarify the question a bit. Consider given a Radon measure $\mu$ on $\mathbb{R}^n$. Let ...
5
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50 views

Masas in quotients

Let $A$ be a von Neumann algebra and let $B$ be a norm-closed ideal of $A$ (but not necessarily WOT-closed). What one has to assume about $A$ and $B$ to ensure that if $M\subset A$ is a maximal ...
5
votes
0answers
158 views

Don't understand this proof of equivalence of weak solutions to PDE

I'm trying to understand the proof that (c) implies (a) here in the following proposition (here, $\mathcal{V} = L^2(0,T;V)$). See the very last line in the image for that part: $$$$ $$$$ I give here ...
5
votes
0answers
102 views

Is this a spectral decomposition/embedding/isometry?

Given a symmetric p.s.d matrix G, we know that a gram matrix/inner-product representation, X exists where $G=X^TX$ and $X=U\lambda^{1/2}$ via the eigen decomposition of $G$. Now if I take the same ...
5
votes
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142 views

Operator completly continuous

For $\lambda>0$, let $v(t)=\lambda \phi(t)$. Consider the BVP consisting of the equation $$u'''=\lambda[f(t,[u-v]^*+\gamma)+M(t)] ,t\in (0,1)$$ and (BC):$u(0)=u'(p)=\int_q^1 w(s)u''(s)=0 ...
5
votes
0answers
87 views

Solution to $\Delta_g u = \delta-1$ on a 2-sphere.

Let $S^2$ be the two-sphere, endowed with a Riemannian metric $g$, such that the volume of the sphere w.r.t. this metric is $4\pi$. Let $a \in S^2$. I am looking for an easy way to prove that the ...
5
votes
0answers
229 views

Problem of Scottish Book

Does anyone know if the problem 50 to Banach written in The Scottish Book is resolved? The problem is: Prove that the integral of denjoy is a Baire functional in the space M ( that is to say, in ...
5
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52 views

Interpretation for the Functional Determinant

Let $S:V \rightarrow V$ be a linear operator on the function space $V$. It is possible to define a functional determinant for $S$ via the zeta function regularization process. In specific we define ...
5
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106 views

What is visualization of gradient flow of a functional?

I don't work on functional analysis but during my study, I faced gradient of a functional. I read its definition, but I can not understand why it is a useful tool? Why if a flow can be written as a ...
5
votes
0answers
112 views

How to prove this element is strictly positive?

Let $A$ be a $C^*\text{-algebra}$ and $A_+$ denote the positive elements. An element $a\in A_+$ is called strictly positive if $\overline{aAa}=A$. Want to prove: if $(e_n)$ is an approximate identity ...
5
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295 views

Show that the function is constant

Let $S^n$ be an $n$-dimentional unit sphere. Consider $f: S^n \longrightarrow R_+$ even continuous function. Denote $$ ...
5
votes
0answers
112 views

Is this function in the Sobolev space $H^{2,-s}(\mathbb{R}^3)$?

I have the function $$f(x)=\frac{e^{iz|x-y|}}{4\pi|x-y|}$$ with $y\in\mathbb{R}^3$ and $\Im z>0$. Let $s>\frac{1}{2}$. Clearly it is not in $H^{2,-s}(\mathbb{R}^3)$ for the singularity of order ...
5
votes
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158 views

Proving that a set is closed in $L^2(\mathbb{R})$

this is my first question here so i hope i don't do anything wrong. Excuse any spelling or grammar mistakes, english isn't my mother tongue. I'm reading this paper for my bachelor thesis and have ...
5
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58 views

$e^{iBt}e^{-iAt}$converges as operator norm

Let $A,B$ be self-adjoint operators on $H$,then we can define the strong limit $$ W=s-\lim_{t\to+\infty}e^{iBt}e^{-iAt} $$ If the limit exsists, then W is called the wave operator, which is ...
5
votes
0answers
96 views

Splitting the action of functionals in duals of Sobolev spaces

Update: After some more thinking and asking I've come to the conclusion that there is no reasonable way to achieve this for all possible $\varphi$ because of the mixed terms. I believe something ...
5
votes
0answers
213 views

Question about the Spectral Theorem for Self Adjoint Operators and Eigenvalues

I have been working through Teschl's book "Mathematical Methods in Quantum Mechanics with Applications to Schrodinger Operators" and I am stuck on a problem in Chapter 3. I am trying to prove that if ...