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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Convergence in weak topology implies convergence in norm topology

In Hilbert space why does convergence in weak topology $x_n$ to $x$ imply that $x_n$ converges to $x$ in norm? Thank you very much for your answers. What if I put a condition on weak convergence ...
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36 views

$W^{s_1 , 2} (\Bbb R^n ) \hookrightarrow W^{s_2,4}(\Bbb R^n )$?

How can I prove that $W^{s_1 , 2} (\Bbb R^n ) \hookrightarrow W^{s_2,4}(\Bbb R^n )$ if $s_1 > s_2 + n/4$ ? $W^{s,p}$ denotes a general Sobolev space for $s =0,1,2,\cdots$. The hook means a ...
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56 views

Transpose of the Shift operators

Let $X = \ell^2$ The operators $\textbf{L}$ and $\textbf{R}$ are defined as $$\textbf{R}x = (0, a_0, a_1...) \;\; \textbf{L}x = (a_1, a_2, a_3...) $$ show that they are the transposes of one another ...
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Why is intersection of kernels of finite linear functional a nontrivial linear subspace?

I am trying to prove that the closure of $S=\{x\in X : ||x||=1\}$ in weak topology is the closure of $B_1(0)$ . I have a doubt about what i am doing is correct and for that i need to know whether the ...
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Bounded functionals on Banach spaces.

Let $(X, \|.\|)$ be a Banach space such that $X \subset C([0,1]) $ For every $r\in \mathbb{Q}\cap[0,1], f\rightarrow f(r)$ defines a bounded linear functional on $X$. Prove that there exists a ...
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84 views

Two norms on $C_b([0,\infty])$

$C_b([0,\infty])$ is the space of all bounded, continuous functions. Let $||f||_a=(\int_{0}^{\infty}e^{-ax}|f(x)|^2)^{\frac{1}{2}}$ First I want to prove that it is a norm on $C_b([0,\infty])$. The ...
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500 views

Bounded integral operators in Functional analysis

Let $K: [0,1] \times \mathbb{R}^n \to \mathbb{C}$ have the properties: $K(x,\cdot) \in L^2(\mathbb{R}^n)$ for all $x\in[0,1]$ For every $f\in L^2(\mathbb{R}^n)$ the function $$ x\mapsto ...
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548 views

minimizer of a function

Consider $1<p<\infty $. Let's define the space: $L_{V}^{p}(-1,1)=\left \{ f:(-1,1)\rightarrow \mathbb{R}:\int_{-1}^{1}\left | f(x) \right |^{p}V(x)dx<\infty \right \}$ Consider the norm: ...
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281 views

(SOLVED) Adjoint of Frechet derivative (involving gradient operator)

I need some help with a problem (a homework/programming exercise) regarding the adjoint operator of the Frechet derivative of an operator. I have the forward operator $ F(a) = L_a ^{-1}f $ where ...
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335 views

Functional analysis summary

Anyone knows a good summary containing the most important definitions and theorems about functional analysis.
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98 views

Continuous operator on $L^\infty$

$1<p<\infty$ and $k\in L^\infty([0,1]^2)$ $(Tf)(s)=\int_{0}^{1}k(s,t)f(t)dt$ I want to show that it is a continuous operator $T:L^p([0,1]->L^p([0,1])$ Proof: What I need to show is that ...
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68 views

Simple question on self-adjoint operators

Let $H$ be a complex Hilbert space, $T\in H'$ and $T=T^*$. Here is where I need help: If $\sigma(T)\subset\{0,1\}$ then $T=T^2$. Using the spectral theorem I know that $\{0,1\} \supset q(\sigma(P)) = ...
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254 views

A continuous embedding.

If $2 \le q \le \infty$ and $2s >n$, then is there a continuous embedding $ H^s (\Bbb R^n ) \hookrightarrow L^q(\Bbb R^n) $ ? Here $H^s$ means a general Sobolev space for $s = 0,1,\cdots$.
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458 views

Proof of normal operator and self-adjoint operator

1) Let $T∈L(V,V)$ be a normal operator. Prove that $||T(v)||=||T^*(v)||$ for every $v∈V$. ($T^*$ is the adjoint of $T$) 2) Let $T$ be an operator on the finite dimensional inner product space ...
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169 views

Understanding topologies in dual and bidual.

I don't know if this is a dumb question but i think i better ask and get my confusion clarified . Talking about topologies in a Vector space , Topologies induced by norms are pretty easy to ...
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352 views

Closure of interior and interior of closure in a topological vector space

If $Y$ is a subset of topological vector space $X$ and is compact and convex show that $\overline{Y^\circ} = \overline{Y}$ and $\overline{Y}^\circ = Y^\circ$. I tried this way but I am not sure: ...
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374 views

Complex conjugate of the Hilbert space

Consider a Hilbert space $H=L^2(\mathbb{R}_+)$, take its conjugate $\overline{H} := \left\{f^{+}, f \in H \right\}$, where $+$ stands for the conjugation. Space $\overline{H}$ is a Hilbert space with ...
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751 views

Functions of bounded variation as the dual of $C([a,b])$

I am trying to understand this proposition about the dual of $C([a,b])$. I would like some help with the following: (1) What does the integral with respect to a function of bounded variation mean? ...
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1answer
109 views

Spectrum of a shift composed with a multiplication operator on a vector valued Banach space

Let us consider the space $L_2(\mathbb{R} \times [0,1]; \mathbb{R}^n)$, i.e functions taking values in $\mathbb{R}^n$ and in $L_2$ . Suppose $T$ is a bounded linear operator defined as follows: ...
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389 views

Strictly convex Inequality in $l^p$

Let $1<p<\infty$. Let $x,y\in l^p$ such that $||x||_p=1$, $||y||_p=1$ and $x\neq y$. Would you help me to show that for any $0<t<1$, $||tx+(1-t)y||_p<1$. My answer : By using ...
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78 views

Existence of only one bounded linear functional.

Given $X$ a linear normed space over $K$ . $I$ be arbitrary indexing set , $\{f_\alpha: \alpha \in I\}\subset X$ and a family $\{c_\alpha: \alpha\in I\} \subset K$, I want to know that there exists ...
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173 views

Norm of linear functional; can we take supremum over dense subset?

If $V \subset H$ are Hilbert spaces and $V$ is dense in $H$, is it true that for $f \in H^*$, $$\lVert f \rVert_{H^*} = \sup_{v \in V} \frac{|f(v)|}{\lVert v \rVert_V}?$$ So I mean can we just take ...
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99 views

Hilbert basis of vector space

Let $V$ a vector space with inner product and $X\subset V$ orthonormal. Prove that exists a Hilbert basis (an orthonormal set of vectors with the property that every vector in $V$ can be written as an ...
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198 views

Open mapping theorem and second category

This seems like a fundamental result but I can not solve it of find an solution: Let $M:X\rightarrow U$ be a bounded linear map between Banach spaces. Show that if the range of M is a set of second ...
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53 views

Closed range for maps between banach spaces? [duplicate]

Possible Duplicate: Question about Fredholm operator This seems to be a standard result but I cannot find the solution. Let $M:X \rightarrow U$ be a bounded linear map between two Banach ...
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251 views

Fact about the orthogonal complement of an subset in pre-Hilbert space

I want to show that if $X$ is a pre-Hilbert space and $A$ is a subset of $X$ with an nonempty interior, then $A^{\perp} = \{ 0 \}$. I tried to assume the contrary, then there would be an $x \ne 0$ ...
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88 views

Does linear isometries always exists?

Given two Banach spaces $X$, $Y$, is always the case that a linear isometry $T: X \to Y$ exists? I actually just want to find a map preserving the unit sphere in infinite-dimensional spaces, and an ...
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133 views

Orthogonal projections on a complete convex set

Let assume it is already known that: If $H$ is an inner product space and $\varnothing \neq A \subset H$ is a complete convex subset, then there exists a unique vector $P_A f:=g\in A$ with $\|f-g\| = ...
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62 views

$T(B_E)$ is closed for $T$ a bounded linear map

Let $T:E\rightarrow F$ be a bounded linear map for E and F Banach spaces, and E reflexive. Let $B_E$ be the unitary closed ball in $E$. How would you argue that $T(B_E)$ is closed?
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Closure in the Space of Probability Measures with the Prohorov metric

I have seen this result stated countless times: assume the metric space $(\theta,d)$ is separable; then $(\theta,d)$ is complete if and only if the space $(\mathcal{P}(\Theta),\rho)$ (the space of ...
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Dual space of the sobolev spaces.

What is the dual space of $ H¹(\Omega) = W^{1,2}(\Omega) $? What is the dual space of $ W^{m,p}(\Omega) $? I know for example that the dual space of $ L^{p}(\Omega) $ for $ 1 \le p < \infty $ is $ ...
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81 views

Determine operator norm of mutiplication operator

Consider $$T: (C[-1,1],\|\cdot\|_{2})\rightarrow \mathbb{C}\\Tf :=\int_{-1}^{1}mf\,\mathrm{d}x$$ where $m\in C[-1,1]$. I want to prove $\|T\| = \|m\|_2$. $\|T\|\leq\|m\|_2$ can be easily proved by ...
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249 views

Show that norm of a functional is continuous

This question is based on Lemma 3.3, page 6 in this paper: http://arxiv.org/pdf/1106.0622v4.pdf I changed the notation quite a lot, but it should be a one-to-one correspondence. $S(x)$ is a compact ...
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Norm in L2 bounded by norm in H1

I am studying FEM the very basics. I don't have a very strong background in math nor in functional analysis. Having said that, here's the problem I'm analyzing. $$ -\mu u'' + \sigma u = f ~~~~~ x \in ...
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358 views

Show that the Volterra operator have dense range.

Let $V: C([0,1]) \rightarrow C([0,1])$ be defined by $$ V f(x) = \int\limits_0^x f(t) dt.$$ Show that V has dense range and find the transpose of V. V has dense range: Since the polynomials are dense ...
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68 views

Showing it is an orthogonal projector

Let $\phi:[0,1]\to\mathbb R$ contionuous and $A:L_2([0,1])\to L_2([0,1])$ defined by $(Af)(x)=\phi(x)\int_{0}^{1}\phi(t)f(t)dt$ I already showed that $A=A^*$ and that $A$ is positive, but I would ...
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163 views

orthogonal projection relation

I try to solve some exercises from Conway "Functional Analysis", and I have some problems with the following: $P,Q,PQ$ are orthogonal projections. (1) $PQ$ is orthogonal projection iff $PQ=QP$ (2) ...
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1answer
146 views

Eigenvalues of Hilbert-Schmith operator

I am having trouble determining the eigenvalues and eigenvectors of the operator $Kv(x)= \int_0^1((x+t)v(t)dt$, where the kernel is $k=x+t$. I have tried to solve the equation $Kv(x)=\lambda v(x)$, ...
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Do $e_1$, $e_2$, … generate the entire $l_2$ space?

From our textbook goes some statement like this: ... let $X$ be the linear subspace of $l_2$ generated by the vectors $$\left\{e_1,e_2,e_3,...\right\}$$ ... Which feels strange to me because I ...
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345 views

Vanishing at the infinity of a function in the Sobolev space.

If $f \in H^s (\Bbb R^n)$ for $s > 1 + \frac{n}{2} $ then the Sobolev inequality implies that $f$ and $\nabla^\alpha f$ ($|\alpha| =1$) vanishes at the infinity. ($\alpha$ : multi-index). But in ...
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proving continuity - interchanging limits

Let $1 \leq p < \infty$ and let $(\alpha_{ij})_{i,j=1}^{\infty}$ be an 'infinite matrix' such that for all $\xi = (x_n) \in \ell^p$ $$(A\xi)_n := \sum_{k=1}^\infty \alpha_{nk}x_k$$ converges and ...
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108 views

Compact Operator on banach espace

Hi I just dont know if this proposition is true, I think it is but I dont know how to start: Let $X$ be a Banach space of infinite dimension, if $T \in B(x)$ and there is an $N$ such that $T^N=I$ ...
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32 views

Finding operator with specific properties

Let $H=(\mathbb R^2,(.,.))$ and $M=\{(x,0)|x\in\mathbb R\}, N=\{(x,x\tan(\theta)|x\in\mathbb R)$ with $\theta\in(0,\frac{\pi}{2})$. Now I would like to find a $T_\theta\in B(H,H)$ with ...
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1answer
188 views

Compact linear operators composition on Banach space implies that one of them is compact?

Hi i just have some question regarding this problem , I cannot find a counter example and neither a proof: Let $X$ be a Banach space of infinite dimension, and $S,T\in B(X)$ (the set of bounded ...
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63 views

How can we show that the following is a norm

Let $X=K^3$. For $x=(x(1),x(2),x(3))\in X$, let $||x||=[(|x(1)|^2+|x(2)|^2)^\frac{3}{2}+|x(3)|^3]^\frac{1}{3}$. Then $||.||$ is a norm on $K^3$
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556 views

Weakly Cauchy sequences need not be weakly convergent

A sequence $(x_n)$ in a Banach space $X$ is called weakly Cauchy if for every $\ell \in X'$ the sequence $(\ell(x_n))$ is Cauchy in the scalar field. I want to show that weakly Cauchy sequences ...
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Must-read papers in Operator Theory

I have basically finished my grad school applications and have some time at hand. I want to start reading some classic papers in Operator Theory so as to breathe more culture here. I have read some ...
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1answer
83 views

Is it ok to switch the limits in $L_2$?

Let $(X,B,\mu)$ be a probability space and let $U$ be a unitary operator on $L_2(X,B,\mu)$. Suppose that $g_n$ is a convergent sequence in $L_2(X,B,\mu)$, $g_n\rightarrow g$. Suppose also that there ...
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189 views

Weak convergence of sequence of polynomials.

A sequence $(x_n)$ in a normed linear space $X$ is said to converge weakly to $x$ if $$ \lim _{n\rightarrow\infty} \ell(x_n) = \ell(x)$$ Consider the sequence $(f_n) \in C([0,1])$ defined by $$f_n(t) ...
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1answer
196 views

Linear extension and Hahn Banach Theorem. Am I missing some detail in this exercise?

This is an exercise problem from a course in functional analysis. However, it is not a homework problem. I think I got it figured out, however my teacher said something during the lecture that I ...