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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1answer
38 views

Is it true $\lambda_i(f(A))=f(\lambda_i(A))$ for non-negative and non-decreasing function on $[0,\infty)$

I am wondering is it next true: Suppose that $f(t)$ is non-negative and non-decreasing function on $[0,\infty)$ and let $A$ be a positive operator on some infinite-dimensional separable Hilbert ...
0
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0answers
36 views

The space of continuous functions as a dual space

Let $X$ be some topological Hausdorff space and $C_b(X)$ the space of bounded complex continuous functions on $X$. Is there a Banach space $B$ such that $B^* \simeq C_b (X)$? I know of a very similar ...
3
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1answer
32 views

Convergence that preserves smoothness

One of the advantages of uniform convergence is that it preserves continuity (among other properties). Unfortunately, it does not preserve derivability. Is there a convergence mode preserving it?
3
votes
1answer
22 views

Weakly compact operator with different domains

Let $A$ be a Banach algebra. Suppose that $e\in A$ such that $e^2=e$ and $eAe$ is division algebra(i.e., $eAe$ is unital and every element of $eAe$ has inverse in $eAe$). Define $T_e:A\to A$ with ...
3
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1answer
28 views

Question about assumptions for Picard-Lindelöf Theorem in Zeidler's functional analysis text

In Zeidler's text on functional analysis pg.24 he wrote... The Picard Lindelöf Theorem: Assume the following: (a) the function $F: S \to \mathbb{R}$ is continuous and the partial derivative ...
2
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1answer
34 views

Help understanding Rudin's proof showing that $C_c(X)$ is dense in $L^p(\mu)$

The proof is from Rudin's "Real and Complex Analysis." It states For $1\leq p<\infty$, $C_c(X)$ is dense in $L^p(\mu)$ The proof is Let $S$ be the class of all complex, measurable, simple ...
2
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0answers
19 views

Why does minimizing $H[f] =\sum^{N}_{i=1}(y_i-f(x_i))^2+\lambda \| Pf \|^2 $ leads to solution of the form $ f(x) =\sum^N_{i=1}c_iG(x; x_i)+p(x)$?

I was reading the following paper of dimensionality reduction (1) and also one on theory of networks for approximations and learning (2) and was trying to understand how the regularization problem ...
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0answers
17 views

finite differences nonlinear least squares

I am facing to following nonlinear least-squares problem: $$\min_{u,\gamma} \frac{1}{1000} \int_{ \gamma(x,y)^2} + \int_{ [u(x,y)−u (x,y)]^2} + \int_{ [∆u(x,y)−\gamma(x,y)u(x,y)]^2}$$ where the ...
3
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1answer
57 views

Prove that ${\{f_n\}}_{n\in\Bbb{N}}$ has a subsequence that converges uniformly to a continuous function on $[0,1]$

Consider the sequence ${\{f_n\}}_{n\in\Bbb{N}}$, where for each $n\in \Bbb{N}$ the function $f_n:[0,1] \to \Bbb{R}$ is absolutely continuous and satisfies $f_n(0)=13$ and $$\int_{[0,1]}|f_n'|^4dx ...
1
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1answer
82 views

Equivalence of holomorphic functions

Given that $$\left(1-\frac{z}{\zeta_j}\right)^{-z}=\sum\limits_{k=1}^\chi\frac{z^k}{k\zeta_j^k},$$ where $\chi$ is the largest nonnegative integer $k$ for which ...
2
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1answer
57 views

The sequence $f_n=x^n$ is not weakly convergent in $C[0,1]$

Let's consider the sequence $f_n=x^n$ for $n \in \mathbb{N}$ in $C[0,1]$ equipped with the usual supremum norm. How can we show that $f_n$ does not converge weakly in $C[0,1]$ without using an ...
0
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1answer
99 views

Weak convergence and convergence almost everywhere

If a bounded sequence $(u_n)$ converge weakly to $u$ in $W^{1,p}(\Omega)$ (Where $\Omega$ is an open bounded from $\mathbb{R}^N$ and $N>p$) Have we that $u_n(x)$ converge to $u(x)$ almost ...
3
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1answer
36 views

What is the $w^{*}$-closure of the finite rank operators in $B(H)$?

I know that the norm closure of the finite rank operators on a Hilbert space is the compact operators $K(H)$. I've been trying to determine what is the $w^{*}$-closure but I am not getting any good ...
1
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0answers
75 views

Alternative ways to prove $\{f:f(0)=\sum_k f(\frac{k}{\sqrt{n}})g_n (k)\}$ is dense in $\{f\in C^2 (\mathbb{R}) : f(0)=\int_{\mathbb{R}} f(u)g(u)du\}$

I want to prove that $$E:=\bigcap_{n\geq 1} \left\{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f\left(\frac{k}{\sqrt{n}}\right)g_n (k)\right\}$$ is a dense subset of: $$F:=\left\{f\in C^2 (\mathbb{R}) ...
2
votes
1answer
32 views

Space between $L^1$ and $BV$?

I am looking for a function space $X_s$ such that this space has following properties: $X_s$ is a Banach space, and has lower semi-continuous properties with respect to $L^p$ strong convergence. I ...
10
votes
2answers
267 views

Algebraically flavoured functional analysis book

I'm looking for a book on functional analysis that would suit someone who is more algebraically/geometrically oriented and seeks to learn the subject with the goal of using it later for geometric ...
0
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1answer
13 views

Set of Riesz homomorphisms

In a text I am using it states the following: "The set of all Riesz homomorphisms between two Riesz spaces does not ordinarily have a simple structure of its own. Consider for example, the set of ...
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0answers
55 views

Operator form $L^2$ space to$L^1$

Can we have an operator such that it transforms an element of $L^2$ to $L^1$? Is this a valid question or this is incorrect? We can consider the measure space as finite.
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0answers
34 views

How to construct examples of functions in the Spaces of type $\mathcal{S}$

We know that there are $3$ types of $\mathcal{S}$-type Spaces, namely $\mathcal{S}_\alpha\:,\: \mathcal{S}^\beta\:,\:\mathcal{S}_\alpha^\beta$. $\mathcal{S}_\alpha: |x^k\varphi^{(q)}(x)|\le ...
1
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1answer
29 views

Please give me an example of closed subspace of banach space under some conditions

Please give me one example of Banach space $X$ and its closed subspaces $S,T,U$ which suffice following conditions. Any of $S+T,T+U,U+S$ is not a closed subspace of $X$. I can say there are some ...
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1answer
54 views

What's the dual of a direct sum? [closed]

I found here two different answers for this question. Which one is it? (I am referring to the dual space of a direct sum). answer1 and answer2
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2answers
37 views

Ordering : Ranges

Given a Hilbert space $\mathcal{H}$. Denote selfadjoints: $$\mathcal{S}(\mathcal{H}):=\{A\in\mathcal{B}(\mathcal{H}):A=A^*\}$$ Note that one has: $$\Delta ...
0
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1answer
40 views

$L^2$ inner product split over sub-domains in $\mathbb{R}^3$

I have a bounded Lipschitz domains $\Omega, \Omega_1, \Omega_2 \subset \mathbb{R}^3$ such that $\overline{\Omega}=\overline{\Omega}_1 \cup \overline{\Omega}_2$ and $\Omega_1 \cap \Omega_2=\emptyset$. ...
0
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2answers
40 views

Can I always write a bounded operator $T$ as $T=R^{*}S$

If $K$ and $H$ are Hilbert spaces and $T\in B(K)$, can I always express $T$ as a linear combination of products $R^{*}S$ for $R,S \in B(K,H)$ ? I think I already showed this is true when $K$ and $H$ ...
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3answers
43 views

Constructing true metrics in infinite dimensional vector spaces?

Is there an example of a true metric defined on a function space? I'd imagine it is some type of integral involving two functions, and it will return a value that obeys the metric axioms, but I have ...
0
votes
2answers
18 views

Can we embed unital Banach algebras into semi-simple ones?

A Banach algebra is (Jacobson) semi-simple if the intersection of all maximal left ideals is the zero ideal. Take a unital abelian Banach algebra $B$. Can we embed it unitally into an abelian ...
0
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0answers
21 views

Is there one to one relation Positive definite(PD) matrix and PD function?

Is it correct to say that a PD matrix can be built from a PD function? For example circulant matrix or Toeplitz seems to be built from a positive definite function. Positive definite function is ...
4
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0answers
30 views

Proof of Sobolev imbedding theorem in Adams

I am struggling to understand the proof of the Sobolev embedding theorem given in Sobolev Spaces by Adams. Specifically section 4.25 (2003 edition). The aim is to prove $W^{m,1}(\Omega) \to ...
0
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0answers
43 views

Convergence of sequence as $O(1/n)$ using damped iterations

I am trying to understand the argument for convergence here (page 16) for damped iterations of non-expansive maps. Say we have a sequence $\{x^n\}_{n=0}^{\infty}$ that is generated as $x^n = \theta ...
2
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0answers
34 views

Convergence of continuation scheme for fixed-point via homotopy

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a non-expansive map, i.e. $$\|f(x) - f(y)\| \leq \|x - y\|$$ for all $x,y\in\mathbb{R}^n$. Further, assume $f$ has at least one fixed-point $x^\star$. ...
3
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0answers
63 views

When weakly compactness implies compactness?

Let $A$ be a Banach space. The weak topology on $A$ is a topology which produced by the following family of seminorms: $~~~~~~~~~~~~~~~~~~~~P_f(x)=|f(x)|,\qquad$ where $f\in A^*$ and $A^*$ is dual ...
2
votes
2answers
47 views

Projections: Orthogonality

Given a unital C*-algebra $1\in\mathcal{A}$. Consider projections: $$P^2=P=P^*\quad P'^2=P'=P'^*$$ Order them by: $$P\perp P':\iff\sigma(\Sigma P)\leq1\quad(\Sigma P:=P+P')$$ Then equivalently: ...
2
votes
4answers
75 views

Are there relations between elements of $L^p$ spaces?

I have read about dual spaces and the relation $1/p+1/q=1$ as mentioned in the Wikipedia page. Are there any more theorems or relations that connect elements between the $L^p$ spaces for different ...
0
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2answers
33 views

Fixed-point analysis similar to Banach Fixed Point Thm

I have a fixed-point question similar to the Banach fixed-point theorem. Let $f\colon \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a given function and let $x^\star \in \mathbb{R}^n$ be a known ...
0
votes
2answers
22 views

How to represent non-linear operators computationally?

I have a finite dimensional vector space V, and want to compute a non-linear operator $R: V \rightarrow V$. I want to have a "general" form of this operator R. I think of the following series ...
1
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2answers
143 views

Lang's treatment of product of Radon measures

Let $X$ be a locally compact Hausdorff space. We denote by $\mathcal B(X)$ the $\sigma$-algebra of Borel sets of $X$. A positive Radon measure $\mu$ on $X$ is a measure defined on $\mathcal B(X)$ with ...
0
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1answer
22 views

Uniform Boundedness: Nets

I thought so far that the uniform boundedness principle applies (according to the proof I know) to any net of bounded operators. Now I read that this works for sequences only. Can you shortly explain ...
0
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1answer
30 views

Summability: Equivalence

Summability Given a Banach space $E$. Consider sums: ...
2
votes
1answer
36 views

Examples of semi-norms on the real line

I just would like to solicit some examples of semi-norms on the set $\mathbb{R}$? Of course, one such examples is the absolute value function. Are there any examples?
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0answers
13 views

4th order correlations of a delta-correlated random process

Say I have a complex random variable A(z) that is $\delta$-correlated, i.e. I have: $ \begin{align}\langle A(z) \rangle &= 0 \\ \langle A(z) A^*(z') \rangle &= \delta(z-z') \end{align} $ ...
2
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2answers
109 views
+400

Spectral theorem for representations proof.

Let $H$ be a separable Hilbert space, and $U$ a unitary representation of $\mathbb{Z}^d$ on $H$. Let $\chi_m$ be the characters of the Torus $T^d$, and $m$ the Haar measure on $T^d$. I would like to ...
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0answers
18 views

Weak harnack type inequality

I have reached a lemma which I do not have any reference and hint for it. Assume $ \Omega \subset \mathbb{R^N} $ is smooth bounded domain and let $u$ be a positive distributional supersolution ...
1
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1answer
24 views

How can we derive the projection formula in general?

The derivation of the well-known projection formula $proj_\vec{b}(\vec{a})=\frac{\vec{a}\cdot \vec{b}}{\vec{b}\cdot \vec{b}}\vec{b}$ uses an argument based completely on geometry. We assume vectors ...
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votes
1answer
35 views

The Frechet derivative can be defined in 2 ways. Is there an easy way of showing that they are equivalent?

$F: X \to Y$ ($X$, $Y$ normed vector spaces) then exists a linear transformation $A:X \to Y$ if for every $\epsilon > 0$ exists $\delta >0$ such that $||F(x+h)-F(x)-Ah||\leq \epsilon ||h||$ for ...
2
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0answers
27 views

$C^0([0,1])$ is separable – or isn‘t it?

Using Bernstein polynoms it can be proven that $(X, \|\cdot\|) := (C^0([0,1]), \|\cdot\|_{C^0([0,1])})$ is a seperable vector space. However, here is my “proof” that this space is not seperable: ...
2
votes
2answers
69 views

Best approximation and an inequality

Let $H$ be a Hilbert space. Let $E\subset H$ and $x\notin E$. Suppose that there exists $y^*\in E$ such that $$\|x-y^*\|=\min_{y\in E}\|x-y\|$$ (i.e., $y^*$ is the best approximant of $x$). I hope ...
0
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0answers
17 views

Uniformly continuous unitary representations.

Let $U$ be a unitary rep. of $\mathbb{R}^d$ on a separable Hilbert space $H$, and $H\cong\oplus L^2_{\mu_v}(\mathbb{R}^d)$ be the spectral decomposition (according to the spectral theorem for these ...
0
votes
0answers
37 views

Dirac Delta Distribution and non-compactly supported test function

I would like to know if there is any problem with defining the following expression: $$ I = \int_0^\infty g(t) \delta(f(t))\mathrm{d}t $$ where $0<\lim\limits_{t\to\infty} g(t) =L<\infty$ and ...
1
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1answer
18 views

Limit inferior, weak convergence

I have a question about weak convergence and limit inferior. Let $(X,\Sigma,\mu)$ be a measure space (if necessary $\sigma$-finite measure space). Let $(u_{t})_{t >0}$ be a family of square ...
1
vote
1answer
38 views

Finding a maximal complete subspace of Riemann Integrable functions on $[0,1]$

I know that the space of Riemann Integrable functions on $[0,1]$ is not complete under the norm $|f|= \int f$. So I was wondering as to what would be a maximal complete subspace of Riemann Integrable ...