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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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 ...
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86 views

Question about a (relatively simple looking) differential operator and its eigenvalues

A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form $L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$ for ...
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56 views

How can I show that given a norm one linear functional on $c_0$ that there is a unique extension to a norm one functional on $\ell_\infty$?

We are given that our Banach space is $c_0 \subset \ell_\infty(\mathbb{N})$ and there is a functional $y^* \in c_0^*$ such that $||y^*|| = 1$. We are guaranteed that this extends, via Hahn-Banach to a ...
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253 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 ...
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140 views

Showing a sequence $x_{n+1} = Tx_n$ forms a contraction mapping

I want to show that the sequence given by $$x_{n+1} = Tx_n = x_n-\frac{(x_n^2-2)}{x_n+x_{n-1}}$$ forms a contraction mapping. That is $$|Tx_1-Tx_2|\leq c|x_1-x_2|.$$ Where $c$ is to be determined. I ...
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158 views

Moduli of smoothness, Besov spaces, and Sobolev spaces

For $1\leq p\leq\infty$, the $r$-th order $L^p$-modulus of smoothness is \begin{equation} \omega_r(u,t,\Omega)_p=\sup_{|h|\leq t}\|\Delta_h^ru\|_{L^p(\Omega_{rh})} \end{equation} where ...
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63 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 ...
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235 views

Invertibility of elements in a Banach algebra

Let $X=L^1\cap L^2$, and $\hat{X}$ be the Banach algebra of the image under Fourier transform of $X$. Then do the unital extension $1\dot{+}\hat{X}$ of $X$ by adding a constant function with the norm ...
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173 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 ...
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120 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 ...
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152 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 ...
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95 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 ...
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261 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 ...
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56 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 ...
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133 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 ...
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480 views

Topics of advanced functional analysis

I did a course in introductory functional analysis and liked it. Now I want to learn more functional analysis with the goal of maybe eventually doing research (phd). I tried to find out what ...
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183 views

How to decompose a representation into direct sum of cyclic representation?

Let $U$ be the bilateral shift operator on $l^2 (\mathbb Z)$, let $T=U+U^*$. How to calculate $\sigma(T)$? And how to show there is no cyclic vector for the action of $C^*(T,I)$. Further how to ...
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121 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 ...
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68 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 ...
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135 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 ...
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74 views

Proving continuity on spaces of distributions?

Let $\mathcal{D}'(\Omega)$ be the space of distributions on an open set $\Omega$, and $\mathcal{E}'(\Omega)$ the compactly supported ones. When you have a linear operator ...
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122 views

Proofing an inequality with a slowly varying function

I am working right now with "Independent and Stationary Sequences of Random Variables" from Ibragimov 1971. I am trying to understand the proof of the following Lemma (18.2.4): $h: \mathbb N ...
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578 views

Distinct eigenvalues for integral operator?

Is there some sufficient condition on the kernel $K$ of a (say, finite rank, to simplify) integral operator $$ \mathcal K:f(x)\in L^2(\mathbb R)\mapsto \int K(x,y)f(y)dy $$ so that it has all its ...
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166 views

Question about proof of Tychonoff-Alaoglu

I'd like to check that I understand the proof in full detail. Can you tell me if the following is correct? Thanks for your help. Claim: The closed unit ball $B_{\|\cdot\|_{op}}(0,1)$ in $X^\ast$ is ...
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2k views

Gradient operator the adjoint of (minus) divergence operator?

Recently I found this statement -- the gradient operator is the adjoint of the minus divergence operator -- in one of my lecture notes. Knowing only a little about functional analysis, I'm looking for ...
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172 views

Distributional derivative of bounded functions

Let $f$ be a bounded measurable function on $\mathbb{R}$. We may consider its derivative $f'$ as a distribution on $\mathbb{R}$. Is there a reasonable description of those distributions $\psi$ ...
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497 views

Eigenprojection as Contour Integral over Resolvent

Let $H$ be a Hilbert space and let $A \in L(H)$ be a bounded linear operator. Assume that $\lambda$ is an eigenvalue of $A$ and assume further that $C_\lambda$ is a simple closed curve in the complex ...
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191 views

Relations between spectrum and quadratic forms in the unbounded case

Let $H$ be a complex Hilbert space. If $B$ is a bounded self-adjoint operator on $H$ then its spectrum is a closed and bounded subset of the real line and we can find its extremes in terms of the ...
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125 views

Direct limits of completely positive maps on $C^*$-algebras vs. operator systems

I believe I've heard, as part of the "lore," that the category (operator systems, completely positive maps) has direct limits, whereas the category ($C^*$-algebras, completely positive maps) does not. ...
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509 views

Are Arzelà–Ascoli theorems results of similar theorems on normed spaces, metric spaces or other spaces?

From Wikipedia, two generalizations of the Arzelà–Ascoli theorem are Let $X$ be a compact Hausdorff space. Then a subset $F$ of $C(X)$, the set of real-valued continuous functions on X, is ...
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129 views

Convergence of integrals of Radon measures

Let $X$ be a locally compact Hausdorff space and let $\mu_n$ be a sequence of bounded variation Radon measures on $X$ such that $\int_X g \;d\mu_n \rightarrow \int_X g \;d\mu$ for each $g \in C_0 (X)$ ...
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331 views

Weakest topology with respect to which ALL linear functionals are continuous

One often considers a Banach space $X$ under the "weak topology", ie. the weakest topology such that all bounded linear functionals are continuous. This leads me to wonder about the weakest topology ...
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205 views

Fractional iteration of the Newton-approximation-formula: how to resolve one unknown parameter?

For my own exercising I tried to find a closed form expression for the Newton-approximation algorithm, beginning with the simple example for getting the squareroot of some given $ \small z^2 $ by ...
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54 views

Null Functional on $l^2$

Let $l^2$ be the hilbert space of all complex sequences $\psi= (\psi_n)_{n=0}^{\infty}$ such that $\sum_{j=0}^{\infty} |\psi_j |^2 < \infty$. Let $\phi= (\phi_n)_{n=0}^{\infty}$ be a sequence of ...
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Function $f:\mathbb{R}^k \rightarrow \mathbb{H}$ is differentiable if partial derivatives are continuous.

Let $f: \mathbb{R}^k \rightarrow \mathbb{H}$ be a function from the euclidean space to a Hilbert space. Is it true that if the partial derivatives are continuous, then $f$ is differentiable? (the ...
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53 views

Relation between weak convergence of probability measures and weak-* convergence

I am trying to nail down the relation between probability and functional analysis. In particular, how the notion of weak convergence used in probability theory is related to the weak-* convergence of ...
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105 views

If $f:X \rightarrow Y$ is a linear isomorphism between $X$ and $f(X)$, then show that there exists a continuous linear map from $Y^*$ onto $X^*$

For any $x \in X$, define the set $\mathcal{F}(X) = \overline{\operatorname{span} \{ \delta_x : x \in X \}}$ where $\delta_x(f)=f(x)$ for all $f \in$ $\operatorname{Lip}_0(X)$. The set ...
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32 views

Help to understand a Lemma about 'supremum of a family of measures'

I read the following Lemma from a paper but I can't understand the proof. Please help! Lemma: Let $\mu$ be a positive measure defined on the family of open subsets of $\Omega$, which is ...
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22 views

Examples of a Banach space with an algebra structure having only left continuity

There is a theorem (see for example, Rudin's Functional Analysis, theorem 10.2 ) that if $A$ is a Banach space with an algebra structure, such that both left and right multiplication are continuous, ...
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53 views

Absolute convergence in Banach space

Let $X$ be a Banach space. Show that, if $\sum_{n\ge 1}x_{n}$ is absolutely convergent then $\sum_{n\ge 1}x_{n}$ is convergent and $$\left\|\sum_{n\ge 1}x_{n}\right\|\le \sum_{n\ge 1}\|x_{n}\|$$ I ...
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64 views

Rigorous proof of this limit

I have shown that the function $$f(x):=\int_{[-\pi,\pi]^n} \frac{e^{-i\langle k,x \rangle}}{1-\frac{1}{n} \sum_{i=1}^n \cos(k_i)}dk$$ exists everywhere for $n \ge 3$. Now, I want to show that ...
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38 views

Fractional Sobolev Space Trace Inequality

Let $\phi\in\mathcal{S}(\mathbb{R}^{n})$ be a Schwartz function, such that ${\phi}\equiv 1$ on the unit ball $|\xi|\leq 1$ and $\text{supp}({\phi})\subset B_{2}(0)$. Set $\phi_{0}=\phi$ and ...
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33 views

Given a function $\phi$ and the set of iterates $(\phi^t)$ under self-composition, can you construct $\phi^t $ for a real continuous parameter?

$\phi$ is given as any function with the following properties: 1. $\phi$ is strictly increasing 2. $\phi$ is continuous 3. $\phi$ is a mapping of the interval $[0,1]$ into itself. 4. ...
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50 views

Why do these Integration-by-Parts Evaluation Terms Vanish?

The Associated Legendre operator is $$ L_mf = -\frac{d}{dx}\left((1-x^{2})\frac{df}{dx}\right)+\frac{m^{2}}{1-x^{2}}f, $$ where $m$ is a positive integer. For the purposes here, define ...
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78 views

Are uncountable “Schauder-like” bases studied/used?

We could define the following notion of basis in a way analogous to unconditional Schauder basis: If $X$ is a topological vector space over $\mathbb R$ and $B=\{b_i; i\in I\}$ be a subset of $X$. ...
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132 views

Definition of a norming set

In this paper, page $131$, in the proof of Proposition $4.1$, there is this sentence: We first note that if $V$ is a separable Banach space, the subset of Lip$(V)$ consisting of all weakly ...
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52 views

(Exponential) Growth of Operator Norm of Uncentered Maximal Function

Define the uncentered Hardy-Littlewood maximal operator $M$ by $$Mf(x):=\sup_{x\in B}\dfrac{1}{\left|B\right|}\int_{B}\left|f\right|,$$ where we the supremum is taken over all (open) balls $B$ ...
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23 views

Continuity on topological dual when the latter is endowed with the weak* topology.

If $\mathbb{E}^*(\mathbb{R}^n)$ is the topological dual of the space o smooth function on $\mathbb{R}^n$ and $\mathbb{S}^*(\mathbb{R}^n)$ is the topological dual of the Schwartz space ...
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62 views

Baker Campbell Hausdorff formula for unbounded operators

Baker Campbell Hausdorff formula says that for elements $X,Y$ of a Lie algebra we have $$e^Xe^Y=\exp(X+Y+\frac12[X,Y]+...),$$ which for $[X,Y]$ being central reduces to ...
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51 views

Bishop-Phelps theorem

Bishop-Phelps Theorem: If $E$ is a Banach space and $B\subseteq E$ is bounded, closed and convex, then the linear functionals on $E$, which attain their supremum on $B$, are norm-dense in $E^*$. ...