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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25 views

$K$ and $L$ homeomorphic, then $C(K)$ is isomorphic to $C(L)$

Can someone sketch the proof (or give me some reference) of the following fact : If $K$ and $L$, compact and Hausdorff spaces, are homeomorphic then the lattices $C(K)$ and $C(L)$ are isomorphic. (I ...
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0answers
60 views

Convexity of $\int f(\frac{dP}{dQ})dQ$ for some convex $f$

I want to prove the convexity of $\int f(\frac{dP}{dQ})dQ$ for some convex $f$ and here is what I've done so far: Since $f$ is convex, $f(\frac{dP}{dQ})$ is also convex w.r.t. $dP$ because $dQ$ ...
2
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1answer
108 views

continuity of $L^p$ norms with respect to $p$

Let $0<p_0<p<p<p_1\leq \infty$. Then I have proved $L^{p_0}(\mu)\cap L^{p_1}(\mu)\subseteq L^{p}(\mu)$. In particular, when $p_0=1$, $p_1=\infty$, I have proved further ...
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1answer
78 views

convolution of two functions and relations with their p-norms

Let $f\in L^p(\mathbb{R})$, $g\in L^1(\mathbb{R})$, $1\leq p< \infty$. Then I have proved the convolution $f\ast g\in L^p(\mathbb{R})$ and $||f\ast g||_p\leq ||f||_p||g||_1$. Does $f\ast g$ ...
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336 views

Problem 21 - Trotter theorem , Reed and Simon

This problem if from Methods of modern mathematical physics I :Functional Analysis, by Reed and Simon: Problem 21: Let $\{A_n\}$ be a sequence of selfadjoint operators on a Hilbert space $H$, and let ...
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1answer
129 views

Square root in Banach algebra

Suppose we are given a unital Banach algebra $A$ and an element $a\in A$ such that the spectrum is a subset of the positive reals $\mathbb{R}_{>0}$. Then by a theorem (see for example W. Rudin ...
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1answer
104 views

Continuity of a map from $\mathbb{C}$ to a Banach algebra

Consider the map from $\mathbb{C}$ to a unital Banach algebra $B$ given by $x \mapsto \exp(xb)$ for $b\in B$. I proved that this map is continuous by using the definition of $\exp(xb)$ as a contour ...
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1answer
54 views

Norm of operator $T_x(f) = f(x)$

Let $X$ be a normed vectorspace and $X'$ be the dual space of $X$. For $x \in X$ we can define $T_x: X' \to \mathbb F$ by $T_x(f) := f(x)$. This is indeed an operator in $X''$. I read that $\| T_x \| ...
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1answer
300 views

How to show the metric space is complete?

the space is the Real line with bounded metric (i.e. $d/(1+d)$, $d$: euclidean). We thought that since nd the space real line with euclidean metric is complete and the bounded metric is smaller than ...
4
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1answer
244 views

Ideal in $C(X)$

Let $X$ be a compact Hausdorff space, and $C(X)$ the space of complex-valued continuous functions with maximum-norm. The following problem is driving me little nuts. I want to show that every closed ...
4
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1answer
88 views

A has an self-adjoint extension

Let $A$ be a symmetric operator satisfying $\langle \phi,A\phi\rangle\geq C\lVert \phi\rVert^{2}$ for all $\phi\in \mathcal{D}(A)$ and some $C\in \mathbb{R}$. Show that the deficiency indiecs are ...
2
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1answer
116 views

How to show the space is totally bounded?

the space the Real line with bounded metric (i.e. d/(1+d), d: euclidean) we know that totally boundedness means that there exists a finite epsilon-net. we first approached to question by directly try ...
5
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1answer
149 views

Is this set convex?

I have been trying to show that the following set is convex, with no luck. I am not even entirely convinced that it is in fact convex. A small hint would be greatly appreciated. For $M>0:$ $$ ...
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1answer
39 views

$L(H)$ and functions belong to predual of this space

Does every W*-continuous linear functional belong to $L^1(H)$? is it true? I cannot understand about it. Please regard me.
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78 views

Prove that $d_{1}$ and $d_{2}$ induce the same topology

Suppose $d_{1}(x,y) = |x-y|$, $d_{2}(x, y) = |\phi(x) - \phi(y)|$, where $\phi(x) = \frac{x}{1 + |x|}$. Prove that $d_{1}$ and $d_{2}$ are metrics on $\mathbb{R}$ which induce the same topology. ...
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0answers
450 views

Textbook for functional analysis in the style of Amann/Escher

most textbooks I've seen so far are not concise enough for my taste and try to give way too much motivation. Or they're written with a too large focus on applications... Rudin wasn't bad contentwise, ...
3
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88 views

There exists function sequence $\{f_{n}\}$ converges to $0$ such that $\{a_{n}f_{n}\}$ not converges to $0$

Let $X$ be the vector space of all complex functions on the unit interval $[0,1]$, topologized by the family of seminorms $$p_{x}(f) = |f(x)|, \quad (0 \le x \le 1).$$ Show that there exists a ...
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2answers
65 views

when H is a separable Hilbert space

I studied when H is a separable Hilbert space, every seminorm is norm;but, if H is not separable, it isnot correct. Is it true? pelese help me
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1answer
89 views

Perpendicular function space

Let $$\mathcal{A} = \left\{ f \in L^{2}(\mathbb{R}) \; : \; \int\limits_{\mathbb{R}} f(x) \, dx = 0, ~\int\limits_{\mathbb{R}} xf(x)\, dx = 0 \right\}, $$ Here $f$ can be complexed valued. Now ...
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2answers
49 views

notation question re: function space

This is a quick notation question: when one writes $X: C[0,\infty) \to \mathbb{R}$, what does that mean exactly? Is $C[0,\infty)$ the space of continuous functions with a domain of $[0,\infty)$ and ...
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0answers
78 views

Orthogonal projection. [duplicate]

I have found this question in a book, but I don't know how to use that $\left\Vert P\right\Vert =1$. Question: If $P\in\mathcal{L}(H)$ is a projection and $\left\Vert P\right\Vert =1$, show that $P$ ...
2
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1answer
82 views

Triangle inequality in product space of normed spaces

Let $(X,||.||_X)$ and $(Y,||.||_Y)$ be normed spaces, then $||(x,y)||:=(||x||_X^p+||y||_Y^p)^{\frac{1}{p}}$ is a norm on $X \times Y$. This is absolutely clear to me, but I have troubles to verify ...
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86 views

$ (\mathbb{R}^3, \|.\|_1) $ and $ (\mathbb{R}^3, \|.\|_\infty) $ cannot be isometric [duplicate]

Can anyone prove that the spaces $ (\mathbb{R}^3, \|.\|_1) $ and $ (\mathbb{R}^3, \|.\|_\infty) $ cannot be isometric? Thanks.
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1answer
130 views

Absolutely convergent sums in Banach spaces

Let's say a sum of elements in a Banach space is absolutely convergent if even the sum of the norms converges, i.e. $\sum_{i=1}^\infty ||x_i|| \le \infty$. This condition implies that the sum of the ...
3
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3answers
84 views

If $\sum_n \|x\|< \infty$, how to show that $\sum x_n$ is convergent in the Hilbert space $H$. [duplicate]

Let $\{x_n\}$ be a sequence in a Hilbert space $H$. If $\sum_n \|x\|< \infty$, how to show that $\sum x_n$ is convergent in $H$? There is no doubt that $x_n \rightarrow 0$ as $n \rightarrow ...
1
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1answer
122 views

Intermediate Analysis Book Suggestions

the title basically says it all. I am looking for real/functional analysis books that would be considered "intermediate level" by that I mean books harder than baby rudin, and easier than big rudin. ...
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1answer
157 views

Duals of Hilbert Subspace

So I am confused about something very basic. I'm going to outline my confusion, and would love if someone could point out when I'm saying something wrong. Let $H$ be a Hilbert space. It's dual $H^*$ ...
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1answer
54 views

Conditions on a sequence of functions to satisfy a certain simple! condition in limit

Let $f_n(x):[0,1]\to [0,1]$ be a sequence of continuous functions such that $f_n(x)\leq 1-x$ and $\int_0^1 f_n(x)\frac{1}{1-x} dx=\frac{1}{n}$. I am interested to know what extra conditions I must ...
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111 views

Using Results in Sobolev Spaces

I have two questions about using results in Sobolev Spaces and a last one on an inequality involving Holder spaces: 1.The Gagliardo-Nirenberg-Sobolev Inequality states the following: Assume $1 \leq p ...
5
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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 ...
4
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3answers
491 views

When do inner products of weakly convergent subsequences converge?

If we have 2 weakly convergent subsequences in $L^2(U)$ (for $U$ some bounded open domain with smooth boundary), $u_k\rightharpoonup u$ and $v_k\rightharpoonup v$, under which conditions do we have ...
3
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2answers
157 views

A problem about Linear Operator

$X$ and $Y$ are Banach Spaces.$ T$ is a linear bounded operator from $X \to Y$. There exists a real number $c$ which is positive, such that for any $y$ belonging to $T(X)$, there exists a $x$ which ...
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2answers
114 views

What is known about this space of parametrised Hilbert spaces?

For each $s \in [0,\infty)$, let $H(s)$ be a Hilbert space. Let us suppose for simplicity that $H(s) = L^2(\Omega_s)$, where $\Omega_s$ is some nice domain that depends on $s$ in a nice way. Define ...
2
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0answers
213 views

$X$ be a reflexive space then show that $X$ is Banach Space and is reflexive in any equivalent norm.

Let $X$ be a reflexive space then show that $X$ is Banach Space and is reflexive in any equivalent norm. ...................................................................... I am trying it in a ...
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1answer
56 views

Can we write $||A - B|| \leq ||A||$?

I am confused with the very basic question related with the matrix norm. Can we write $||A - B|| \leq ||A||$ ? Thanks for the help and time.
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1answer
72 views

Showing this operator is densely defined

This is an example in Rudin's Functional Analysis, in the chapter on Unbounded Operators. Consider the right shift operator $V$ on $l^2$. It is an isometry and closed, and $I-V$ is one-one and so $V$ ...
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1answer
236 views

Prove that the infimum is not attained for a set $M$

Consider $C([0,1])$ with the $\sup$-norm. Let $$N = \bigg\{ f\in C([0,1]) | \int_0^1 f(x)dx = 0\bigg\}$$ be the closed linear subspace of $C([0,1])$ of functions with zero mean. Let $$X = \{ f\in ...
1
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1answer
47 views

Norm of a vector is determined by evaluation of linear functionals on it: can this be proved without the Hahn-Banach theorem?

Let $V$ be a normed vector space over the field of real numbers, $\mathbb R$, and let $x_0 \in V$ be fixed. I know how to prove $$\|x_0\| = \sup_{f \in V^*, \|f\| = 1} |f(x_0)|$$ using the Hahn-Banach ...
5
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1answer
109 views

What is Newton's theorem?

I'm reading a paper about mathematical physics at the moment and am wondering about the following: Let $w\colon\mathbb{R}^2\to\mathbb{R}$ be defined by $w(x)=-\log|x|$ and ...
2
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1answer
49 views

Continous and Discrete basis, Multiplication of Density Matrix and Hamiltonian.

Suppose I have a wave function $\psi(x)$ in position basis. I can make a density function by simply multiplying $\psi(x)$ and its conjugate $\psi^*(x)$. If I operate the density matrix ...
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47 views

$X$ be a normed space and assume that $E \subset X$ such that $\operatorname{int}(E) \neq\varnothing$

Let $X$ be a normed space and assume that $E \subset X$ such that $\operatorname{int}(E) \neq \varnothing$ then show that $E$ spans $X$. I am trying it in a following way.... Let be the norm ...
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1answer
611 views

Dual of the space of all convergent sequences

I need to find what it wrong with my logic and Ii will be glad if someone can told me what I do wrong. Define $C$ be the subspace of $ l^{\infty} $ that consists of convergent sequences and let ...
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1answer
132 views

Compact Operators and Complete Metrics Spaces

I have a couple of questions about compact operators and compactness in complete metric spaces: 1.I have the following implications: Let $Y$ be a metric space with $A$ a subset of $Y$. $A$ is ...
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2answers
93 views

What is the closure of $ C^\infty_c(\mathbb{R}^n\setminus\{0\})$ in Sobolev $ W^{1,p} $ norm?

For $1 \leq p < \infty, n\geq 1 $ my guess of the answer was $ W^{1,p}(\mathbb{R}^n)$ but I can't prove the inclusion $ \overline{C^\infty_c(\mathbb{R}^n\setminus\{0\})} \subseteq ...
6
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1answer
107 views

Show that an unbouned normal operator is closed

A linear operator $A$ is called nomal if $\mathcal{D}(A)=\mathcal{D}(A^{*})$ and $\lVert A\phi\rVert =\lVert A^{*}\phi\rVert$ for every $\phi\in \mathcal{D}(A)$. Show that normal operators are closed. ...
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1answer
57 views

Find an idempotent operator with $\ell_\infty$ as its range?

I'm not sure how to show the following question: Let ${X}$ be a Banach space that contains $\ell_\infty$ as a closed subspace. Prove there exists an idempotent bounded linear functional $E$ on ${X}$ ...
2
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1answer
151 views

Is $L^2(\Omega)$ the only $L^p$ hilbertian space?

I've started today studying Hilbertian spaces, and all of the examples seen in class were about the space $L^2(\Omega)$, where $\Omega$ is a limited domain in $\mathbb{R}^N$ $(N \geq 1)$. Online I ...
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1answer
73 views

Showing that there is always a discontinuous functional?

I am supposed to prove the following: Let $(V,||.||)$ be an infinite-dimensional space, then there is always a discontinuous function $T:V \rightarrow \mathbb{K}$ Since continuous is equivalent to ...
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2answers
139 views

Prove that if $\lambda$ is an isolated eigen-value of $T=T^*$, then $\ker(T-\lambda)=E_{\{\lambda\}}H$

Here we have a self-adjoint, densely-defined operator $T$ on a Hilbert space $H$, and $E_M$ is the usual spectral projector for any Borel set $M$, i.e., $E_M=\int_M\text{d}E_t$ (this means, by ...
5
votes
1answer
509 views

Norm of Fredholm integral operator equals norm of its kernel?

Let $T_k(f)(s):=\int_0^1 k(s,t) f(t) dt $, where $k \in L^2([0,1]^2)$ and $f \in L^2([0,1])$. Then it was fairly easy to see that $||T_k|| \le ||k||_{L^2}$, but now I was wondering how to show that ...