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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72 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}) ...
8
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1answer
298 views

Heat Kernel Property

Let $\phi$ be the Heat Kernel in $\mathbb{R}^n$, i. e. $$\phi (x,t)={(4\pi t)}^{-n/2}\exp\left( - \frac{\mid x \mid ^2}{4t}\right)$$ and let $u$ satisfy Heat equation. Show that: ...
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27 views

Statement about the discrete (metric) space, and both an open and closed ball.

I have the following statement from my notes: "Let $(X,d)$ be the discrete space i.e. any non-empty set with the discrete metric ($d_d(x,y)=1$ for all $x\neq y$). Then, amazingly, $B_1(x)=\{x\}$, a ...
3
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2answers
45 views

Prove that the linear transformations are the same.

I have this lemma: If X is a complex inner product space and $S,T \in B(X)$ are such that $(Sz,z)=(Tz,z)\forall z \in X$, then $S=T$. $B(x)$ is the set of bounded linear operators from X to X. ...
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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 $H$ ...
3
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1answer
31 views

Show that there exists a sequence of functions $\{f_n:[0,1]\to\mathbb C\}$ satisfying the given condition.

Show that there exists a sequence of functions $\{f_n:[0,1]\to\mathbb C\}$ satisfying: 1) $f_n\to0$ pointwise; 2) $\gamma_nf_n\not\to0$, for all $\gamma_n\in\mathbb C$ such that ...
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1answer
28 views

Prob. 3, Sec. 3.2 in Kreyszig's Functional Analysis Book: Is the space of all polynomials of a fixed degree complete? [duplicate]

Let $n$ be a given natural number, and let $X$ denote the vector space consisting of the zero polynomial and of all polynomials of degree at most $n$, with real or complex numbers as co-efficients, ...
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30 views

Is this function uniformly convex?

I am using the following definition of uniformly convex: A continuous functional $G:Y \to R$ is uniformly convex on a ball $$B(0,\delta):=\{y\in Y : \|y\| < \delta\}$$ if there exists a ...
2
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1answer
29 views

One norm is stronger than but not equivalent to another norm

Let $X=C^{1}([0,1])$. For $x \in X$, let $$||x||_{0}^{'}=|x(0)|+||x'||_{\infty}$$ I need to show that $||.||_{0}^{'}$ is stronger than but not equivalent $||.||_{\infty}$. It is easy to see that ...
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17 views

Showing the space of $\beta$-operators between Banach lattices is a Banach space [on hold]

Let $E,F$ be a Banach lattices. A linear map $T:E \rightarrow F$ is called $\beta$-operator if $\lVert T\rVert_\beta < \infty$, where $$\lVert T\rVert_\beta := \sup \bigl\{ \bigl\Vert ...
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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?
4
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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 ...
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2answers
265 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 ...
3
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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
19 views

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

In Zeidler's text on functional analysis pg.24 he wrote... The Picard Lindelof Theorem: Assume the following: (a) the function $F: S \to \mathbb{R}$ is continuous and the partial derivative ...
2
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0answers
33 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 $x^\star$. ...
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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 ...
5
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1answer
631 views

Strict Inequality in Rudin's Proof of the Riesz Representation Theorem

In Rudin's proof of the Riesz Representation Theorem (step ten), he proves that $$\Lambda h_i \leq \mu(V_i) < \mu(E_i) + \epsilon/n , \quad \mu(K) \leq \sum\limits_{1 \leq i \leq n} \Lambda h_i.$$ ...
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 ...
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
16 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 ...
2
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1answer
56 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 ...
2
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1answer
71 views

Closure of the Hamilton's operator $(Hf)(x)=\frac{1}{2}f''(x)-V(x)f(x)$ with $C_c^\infty(\mathbb{R}, \mathbb{C})$ domain

Let $V \in C_{b}^{1}(\mathbb{R}, \mathbb{R})$ be a differentiable function bounded with its first derivative and $H$ be a Hamilton's operator such that: $$(Hf)(x)=\frac{1}{2}f''(x)-V(x)f(x)$$ The ...
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1answer
46 views

Nonexpansive Affine Operators in Hilbert spaces

Let $H$ be a Hilbert space with real inner product $\langle \cdot, \cdot \rangle : H \times H \rightarrow \mathbb{R}$. Let $T$ be an affine operator. Show that $T$ is nonexpansive, i.e., $\left\| ...
3
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1answer
29 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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1answer
56 views

Hilbert space inequality $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$

In prelim prep I came across 'given $\epsilon$ there exists $C_{\epsilon}$ such that $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$. It is asserted without proof, so I've tried ...
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1answer
66 views

Norm of two operators

(1) $U(x)=a \langle x,b \rangle +b \langle x,a \rangle $, $a,b\in H\setminus \{0\}$. U is an operator from H to H and a,b are orthogonal elements. I want to calculate $||U||$ For this one I tried the ...
1
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1answer
24 views

Biorthogonal complement of subspace of subspace.

I'm taking a course on Banach and Hilbert spaces. The teacher who guides the exercise sessions is often a bit fast, so only when revising my notes at home I realize I do not fully understand them. We ...
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136 views
+50

Regularity of a weak solution

The problem is formulated as follows: Let $\Omega\subset\mathbb{R}^2$ a bounded Lipschitz domain. For $u\in L_2(\Omega)$ one can define the one dimensional weak dirivative $\partial_1 u$. Now define ...
2
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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
3answers
457 views

Infinite sum of floor functions

I need to compute this (convergent) sum $$\sum_{j=0}^\infty\left(j-2^k\left\lfloor\frac{j}{2^k}\right\rfloor\right)(1-\alpha)^j\alpha$$ But I have no idea how to get rid of the floor thing. I thought ...
2
votes
1answer
31 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 ...
3
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0answers
33 views

The best constant in Poincare-liked inequality in BV space

Let $\Omega\subset \mathbb R^N$ open bounded smooth boundary. Then we could prove that for any $u\in BV(\Omega)$ and $\omega\in \operatorname{ker}\mathcal E $, where $\mathcal E$ denotes the ...
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2answers
75 views

Functional analysis: Why $|[x,y]|\leq [x,x]^{1/2}[y,y]^{1/2}$

Let $\varepsilon>0$. We denote $[x,y]:=\langle \varepsilon x+Ax,y\rangle$ where $A$ is symmetric and $A\geq 0$. Why do we have that $$\big|[x,y]\big|\leq [x,x]^{1/2}[y,y]^{1/2}\ \ \ ?$$ The hint ...
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1answer
42 views

Functional analysis: $\|A\|=\sup\{|\langle Ax,x\rangle|\mid x\in X, \|x\|\leq 1\}$

Let $(X,\langle\cdot ,\cdot\rangle)$ an inner product space and $A\in\mathcal L(X)$. I have to show that $$\|A\|=\sup\{|\langle Ax,x\rangle|\mid x\in X, \|x\|\leq 1\}.$$ The fact that $\|A\|\geq ...
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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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 ...
2
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4answers
74 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 ...
4
votes
5answers
73 views

Properties of $L^2(-1,1)$ functions

I want to show that there is no function $v \in L^2(-1,1)$ with $\int_{-1}^{1} v(x)\phi(x) dx = 2\phi(0)$ for all $\phi \in C^\infty_0(-1, 1)$ ($\phi$ is $0$ everywhere but $[-1,1] $). I know about ...
7
votes
1answer
112 views

Essential Selfadjointness of Quantum Harmonic Oscillator Hamiltonian

The Hamiltonian for the Quantum Harmonic Oscillator is (disregarding constants) the Hermite operator $$ Hf = -f''+x^{2}f, $$ where $\mathcal{D}(H)$ consists of all twice absolutely ...
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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 ...
1
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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.
1
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1answer
28 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 ...
12
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3answers
1k views

Does every $\mathbb{R},\mathbb{C}$ vector space have a norm?

Is there a canonical way to define on any vector space over $\mathbb{K}=\mathbb{R},\mathbb{C}$ a norm ? (Or, if there isn't, can someone give me an example of a vector space over $\mathbb{K}$ that is ...
3
votes
1answer
361 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 ...
-1
votes
1answer
53 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
-1
votes
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 ...
1
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2answers
118 views

Example of a net in $\mathcal{B}(\ell_2)$ that converges in the weak operator topology but not in the strong operator topology?

I need to show that the Strong Operator topology is strictly stronger in $\ell_2$ (space of complex sequences that are square summable). I can show that convergence in the strong operator topology ...
0
votes
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 ...