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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57 views

The continuous dual of the reals

I just have a few questions involving the continuous dual of $\mathbb{R}^{N}$. We know that the dual $(\mathbb{R}^{N})^{*}$ of $\mathbb{R}^{N}$ is the space of all linear forms $$a: \mathbb{R}^{N} \...
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72 views

Hahn-Banach separation theorem

Let $V_1,V_2$ be convex subsets of a normed space $X$ with $V_1^\circ\neq\emptyset$ and $V_1^\circ\cap V_2 =\emptyset$. Then there exists $x'\in X'\setminus\{0\}$ such that $$\operatorname{Re} x'(v_1)\...
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52 views

What's the name of this type operator?

If $H$, is a seperable Hilberspace, $E$ seperable Banach space. $(h_n)$ orthonormal basis of $H$. Let $T\in B(H,E)$. If we have the condition on $T$ that $$\sum_k \left\|Th_k\right\|^p <\infty,$$ ...
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30 views

Modifying a Density Function

Assuming a real an continuous function $f_1(x)$ defined on $\mathbb{R}^+$ which satisfies Probability Density criteria: $$ f_1(x) \geq 0 \quad \forall x \geq 0, \quad \int\limits_0^{+\infty}f_1(x)\...
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84 views

Theorems on continuous embedding

Let $X,Y$ be two normed spaces such that $X\hookrightarrow Y$. Show the following: 1) If $A\subset X$ is closed in $Y$ then $A$ is closed in $X$; 2) If $X, Y$ are Banach spaces if $x_n$ tends ...
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36 views

Estimates in Hölder spaces

Let $u,v\in C^{2,\alpha}\left(\overline{\Omega}\right)$. Proof that there exists a constant $C>0$ so that \begin{equation} \|\Delta v\left(|\nabla v|^2-|\nabla u|^2\right)\|_0\leq C\left(\|u\|_2+\|...
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85 views

Cauchy-Schwarz type formula for positive integral operator

Let $\gamma(x,y)$ be some complex valued function in $L^{2}(\mathbb{R}^{2})$ such that $$ \gamma(x,y)=\overline{\gamma(y,x)},\forall x,y\in \mathbb{R} $$ Let $S=(1-\Delta)^{1/2}$ acting on $\gamma$. ...
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28 views

reverse-reverse of Michael selection theorem

Let $X\subseteq\mathbb R^d$ be a compact and $Y=\mathbb R^d.$ Let $\Gamma:X\twoheadrightarrow Y$ be a multi-valued map with closed values. Assume that $\Gamma$ admits a continuous (single-valued) ...
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1answer
82 views

Spectrum of the operator $A(f,g)=(g,\Delta f-f)$

Let $\Omega$ be an open set in $\mathbb{R^n}$. We consider the product Hilbert space $H=H^1_0(\Omega)\times L^2(\Omega)$ with the norm $$|(f,g)|^2=\int_\Omega (|\nabla f|^2+|f|^2+|g|^2 ) dx$$ We ...
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38 views

Various convergences in the space of bounded operators

Could you please help me to find some classical (counter)examples in functional analysis? Let $X$ and $Y$ be some normed spaces over $\mathbb{C}$. By $\mathcal{B}(X,Y)$ we denote the space of bounded ...
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32 views

Analog of eigenvalue bound for a general bounded operator

It's known that for a matrix, $\max|λ|≤\sqrt{tr(A^*A)}=\sqrt{∑_{i,j=1}^n|A_{i,j}|^2}$ where $\lambda$ denotes its eigenvalue. I'm wondering whether there's an analog of this inequality for a general ...
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169 views

Riesz Lemma for reflexive spaces

I know the proof of Riesz Lemma: Let $Y$ be a closed (proper) subspace of a normed space $X$. Let $\varepsilon >0$. Then it exists an element $x \in X$ such that $||x||=1$ and $d(x, Y) \geq 1-\...
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1answer
171 views

An example of a separable Banach sequence space in which the finite support sequences are not dense?

I am wondering if there exist examples of Banach (or Frechet) sequence spaces in which the set of all finite support sequences are NOT dense?
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39 views

Local Module Homomorphism

Let $A$ be Banach algebra and $X,Y$ be two left Banach $A$-modules. It is said that the linear bounded map $\phi:X\to Y$ is left $A$-module homomorphism if for any $a\in A$ and any $x\in X$ we have $\...
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1answer
70 views

Is $\langle f,g\rangle$ defined for distributions $f,g$?

Consider a standard setting for the development of the theory of distributions. Let $D(\Omega)$ be the space of test functions and $D'(\Omega)$ be the space of distributions ("generalized functions"). ...
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39 views

Closure of a von Neumann bounded set

Let $V$ be a topological vector space and $B \subseteq V$ bounded. Then the closure $\overline{B}$ is bounded. This appears on the Wikipedia page http://en.wikipedia.org/wiki/Bounded_set_(...
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151 views

Unique fixed point of a contraction defined on a closed ball which maps the boundary back into the ball

Let $X$ be a Banach space, $r > 0$, $A: K_r(X) \rightarrow X$ a contraction (where $K_r(X)$ is the closed ball of radius $r$ and center $0$ in $X$), with contraction constant $0<q<1$, which ...
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1answer
262 views

An application of Banach fixed point theorem for initial value problem

Find a condition for $\beta>0$ which implies that the differential equation system \begin{align} x'(t)&=x(t)+y(t) ,\\ y'(t)&=t^{2}+tx(t) \end{align} with initial conditions $x(0)=0, y(0)=...
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0answers
69 views

Fixed-Points Limit Set

Let $f : \mathbb{R}^n \rightarrow X$ be continuous and $X \subset \mathbb{R}^n$ be compact and convex. For all $p \in \mathbb{R}$, consider $A_p \in \mathbb{R}^{n \times n}$, with $p \mapsto A_p$ ...
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174 views

Weakly compact closed balls in reflexive space

If it is given that $X$ is a reflexive Banach space. Let $K \subset X$ be a norm closed and norm bounded convex set. I want to show that $K$ is weakly compact. I have the following idea but I am not ...
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1answer
187 views

Extension of a linear operator

Let $T$ be a linear operator defined on the space of the algebraic polinomials in $[0,1]$ (polinomials with rational coefficients) such that for each $k \in \mathbb{N}, T[x^k]=0$. Is it possibile to ...
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1answer
55 views

Conicide of $w^*$ and norm topology on $S_{\ell_1}$

I want to show that on $S_{\ell_1}=\{x\in \ell_1: ||x||=1\}$the $w^*$-and the norm topologies are coincide. Can any one help me . Thanks
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68 views

What is the definition of regular operator?

If $T$ is a bounded linear operator on a normed space $X$. What "$T$ is regular operator" means?
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134 views

A decomposition of Hilbert space via self-adjoint operator

Let $H$ be a complex Hilbert space and $A:H\to H$ self-adjoint. Show that one can decompose $H$ into two $A$-invariant closed subspaces as $H=H_{p} \bigoplus H_{c}$ such that the spectrum of $A|_{H_{...
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90 views

On maximal ideal spaces of a banach algebra

I am reading this article on maximal ideal spaces and there is this part that I don't quite understand very well, hope you guys can help me out. "Let $M(A)$ denote the maximal ideal space of a ...
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26 views

Question about convergence of sum

Let $T\in B(H,E)$ where $H$ a seperable hilbertspace, $E$ a seperable Banach space. By parsevals identity $$\left\|T^*\right\|^2= \sup_{ \left\|x^*\right\|\leq 1}\left\|T^*x^*\right\|^2 = \sup_{ \...
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1answer
47 views

Bound on number of mutually orthonormal eigenfunctions

Let $E$ be the vector space of real valued continuous functions on an interval $[a,b]$. Let $K = K(x,y)$ be a continuous function of two variables, defined on the square $a \leq x \leq b$ and $a \leq ...
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1answer
148 views

comparing largest eigenvalue of two positive matrices

I have a conjecture that for any two positive matrices(all elements are positive, nothing about positive definite or symmetry) $A$ and $B$, if $A_{ij} A_{ji}>B_{ij} B_{ji}$, while there is certain ...
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51 views

Homology of smooth loop spaces of spheres

I'm looking for a reference on the homology of $C^\infty(S^1,S^n)$. So far I've only been able to find references for spaces of maps which are $C^k$ or in a Sobolev class. I also have references which ...
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199 views

Topological Vector Space: $\dim Z\text{ finite}\implies Z\text{ closed}$

Let $V$ be a Hausdorff topological vector space and $Z$ a linear subspace: $Z\leq X$ Is there a neat way to prove that: $$\dim Z\text{ finite}\implies Z\text{ closed}$$
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1answer
47 views

inductive limit

Consider spaces $$E_n=\{x=\{x_k\}_{k\in\mathbb{N}}\mid x_j=0,\quad j>n\},\quad x_k\in\mathbb{R}$$ endowed with $\|\cdot\|_\infty$ norms. Let $E$ be an inductive limit of these spaces. This set ...
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1answer
69 views

Left shift operator $L: l^2 \rightarrow l^2$ on the sequence space $l^2$

$$L: l^2 \rightarrow l^2$$ is defined by $$b = (b_1,b_2,...) \mapsto Lb = (b_2,b_3,...)$$. $(Lb)_n = b_{n+1}$ respectively. How can I determine the adjoint endomorphism $L^*$? Kind regards George
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101 views

Skew adjoint operator with uncountable spectrum

Let $H$ be a Hilbert space. I just want an example of a skew adjoint operator $(A^*=-A)$ with uncountable spectrum. I also want an example for unbounded differential operators. The only example I ...
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1answer
444 views

An application of Banach fixed theorem on an integral equation

I'm learning some applications of the Banach Fixed Point Theorem and I have the following question: Consider the integral equation $\displaystyle x(t)=\int_{0}^{\frac{\pi}{2}}\arctan \left(\frac{x(s)}...
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1answer
91 views

Identifying the dual group of a locally compact abelian group with the spectrum of $ {L^{1}}(G) $.

Folland stated the following theorem in his book A Course in Abstract Harmonic Analysis on Page 88. The dual group of a locally compact abelian group can be identified with the spectrum of $ {L^{1}...
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1answer
85 views

$\oint_{C}(A-\lambda I)^{-1}\,d\lambda=0$ implies interior of $C$ is in the resolvent.

Suppose that $A$ is a bounded linear operator on a complex Banach space $X$ with resolvent set $\rho(A)$. If $C$ is a simple closed smooth curve in $\rho(A)$ such that $$ \oint_{C}(\...
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1answer
59 views

Is this subspace dense in $L^{2}(\Omega,\mu)$

Let $(\Omega,\mu)$ be a measure space, and let $X=L^{2}(\Omega,\mu)$ be the complex Hilbert space of square-integrable complex measurable functions on $\Omega$. (Each $f \in L^{2}$ is an equivalence ...
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1answer
88 views

Show that $\|e^{tA}\| \le e^{t\|\Re (A)\|}$

Let $X$ be a complex Hilbert space, and let $A$ be a bounded linear operator on $X$. Define the real part of $A$ to be $\Re(A)=\frac{1}{2}(A^{\star}+A)$, and define $e^{tA}=\sum_{n=0}^{\infty}\frac{1}{...
4
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1answer
49 views

Is a function $f$ with $f(X)\perp (I-f)(X)$ necessarily linear?

Let $X$ be a real or complex inner-product space, and let $f : X\rightarrow X$ be a function such that every element of $f(X)$ is orthogonal to every element of $(I-f)(X)$. Prove or give a ...
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1answer
34 views

Continuity in the Strong vs Operator topologies on a compact space

Suppose $X$ is a compact space, $H$ is a Hilbert space, and $f:X \rightarrow B(H)$ is continuous when $B(H)$ is given the strong topology. Does this imply that $f$ is continuous when $B(H)$ is given ...
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48 views

A second question about a proof of Banach-Alaoglu

I have another question about the proof of Banach-Alaoglu using nets. The proof proceeds by considering a universal net into the closed unit ball of $X^\ast$, let's call the ball $S$ and the net $\...
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1answer
43 views

estimate to show bilinear form continouus

Consider the bilinear form $$a(u,v)=2\mu\int_\Omega \mathrm{trace}\left(\varepsilon(u)^T\varepsilon(v)\right)dx + \lambda\int_\Omega \mathrm{div}(u)\mathrm{div}(v)dx$$ with $\varepsilon(u)=\frac{1}{2}...
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40 views

Differentiable structure on the gauge group?

In this paper I have come across a formulation involving differentiation in the gauge group of a principal bundle which I do not understand (found at the very top of p. 369). Let $P\rightarrow M$ be ...
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2answers
48 views

Question about a proof of Banach Alaoglu

I'm reading the proof of Banach Alaoglu using nets. The theorem states that the closed unit ball in $X^\ast$ is weak star compact. My question is: If $\varphi_\alpha$ is a universal net mapping into ...
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0answers
25 views

Characterisation of open mappings

Let $X, Y$ be normed spaces and $F$ - linear continuous mapping. Is it true that $F:X\rightarrow Y$ is open, $\iff F(B(0,\epsilon))$ is open for any $B(0,\epsilon)$ denoting an open ball with radius $...
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1answer
17 views

Characters only on commutative unital algebras?

I saw the following definition of a character: Let $A$ be a commutative unital Banach algebra. Then a non-zero homomorphism $\chi : A \to \mathbb C$ is called a character. For this definition to ...
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1answer
36 views

Question about spectra in non-unital algebras

Let $A$ be a non-unital complex commutative Banach algebra. I am in the process of showing: $$ \sigma (a) = \{\chi (a) \mid \chi \in \Omega (A) \} \cup \{0\}$$ where $\sigma$ denotes the spectrum ...
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1answer
81 views

$L$-Lipschitz gradient of $f$ implies inverse strongly monotone of $\nabla f$

Let $f:\; \mathcal{H} \to R$ be a continuously differentiable convex function such that $$\|\nabla f(x) -\nabla f(y)\|\leq L\|x-y\|.$$ Prove that the mapping $\nabla f$ is $1/L$ inverse strongly ...
3
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1answer
43 views

Subspace of certain series in a Hilbert space is compact

Let $E$ be a Hilbert space and let $\{x_{n}\}$ be an orthonormal basis.  Let $\{c_{n}\}$ be a sequence of positive numbers such that $\sum c_{n}^{2}$ converges.  Let $C$ be the subset of $E$ ...
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2answers
158 views

Does the shift operator on $\ell^2(\mathbb{Z})$ have a logarithm?

Consider the Hilbert space $\ell^2(\mathbb{Z})$, i.e., the space of all sequences $\ldots,a_{-2},a_{-1},a_0,a_1,a_2,\ldots$ of complex numbers such that $\sum_n |a_n|^2 < \infty$ with the usual ...