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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146 views

Span of a closed subspace

Let $Y$ be a closed set of a Banach Space $X$. Is it true that the linear Span($Y$)is also closed? For the examples I have tried, I see that the result holds true. I understand that the linear span ...
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1answer
40 views

Inner product and norm

I have to show that if $\|x+y\|=\|x\|$ then $2x+y$ and $y$ are orthogonal. I think I can use Pythagoras. Thank for any help. Corrected
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2answers
176 views

Proving that the function f is constant, mean value theorem, derivatives

Having the following inequality, for a real-valued function $f$ which is twice differentiable: $f(a+h)-f(a)\geq f(a)-f(a-h)$ for any $a \in\mathbf{R}$, $h > 0$. and assuming that $f$ is bounded, ...
1
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1answer
52 views

Functional defined on the space of functions with compact support

Let $X=C_c(\mathbb{R})$, the space of functions with compact support, normed with the max norm. Define $\Gamma: X \rightarrow \mathbb{R}$ as: $$ \Gamma f= \int_{- \infty}^{\infty} f(t)dt \>\>\&...
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1answer
58 views

Reference for: $L^p(S\times\Omega)$ and $L^p(S;L^p(\Omega))$, $p\in[1,\infty)$, are isometric isomorph.

I am having trouble finding a reference for the following result: Theorem 1. Let $S=(0,T)$ be a finite intervall and $\Omega\subset\mathbb{R}^n$, $n\in\mathbb{N}$, be a bounded domain. Then the ...
2
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1answer
172 views

Cayley Transform: well defined?

Why is the Cayley backtransformation well-defined: $$A_U:=\imath(1+U)(1-U)^{-1}$$ In general $1-U$ is not invertible for example for $U=1$.
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0answers
58 views

Compactness hypothesis in Riesz representation theorem

Let $X$ a compact metric space; I have to identify the dual of the set of continuous functions on $X$, $C(X)^*$. By Riesz representation theorem we have that it can be identified with the space of ...
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1answer
58 views

Eigenvectors of a Multiplication Operator

Let $\mu$ be a positive finite Borel measure on $[-\pi,\pi)$. Define the multiplication operator $M_\mu : L_2(\mu) \to L_2(\mu)$ by $(M_\mu f)(\theta) := e^{i \theta} f(\theta)$. I've proved that $M_\...
4
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2answers
642 views

Matrices A+B=AB implies A commutes with B

$A$ and $B$ are $n\times n$ matrices and $A+B=AB$. I have an interesting proof that this implies $A$ commutes with $B$, but the proof only works when $||B|| \lt 1$. I'm looking for a way to salvage ...
1
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0answers
41 views

Dual space of $L^2(0,T;H^1) + L^p(0,T;L^p)$ and its duality pairing?

Let $V=L^2(0,T;H^1) + L^p(0,T;L^p)$. We know that its dual space is $V^* = L^2(0,T;H^{-1}) \cap L^p(0,T;L^p)$. So if $v \in V$, then by definition $v=a+b$ where $a \in L^2(0,T;H^1)$ and $b \in L^p(0,...
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1answer
44 views

Show that a sequence is weakly convergent to $0$ in $\ell^{2}$

I have to show that a sequence $(f_n)=2009e_n+e_{2010n}$ is weakly convergent to $0$ in $\ell_{2}$ where $e_n$ are standard basis vectors in $\mathbb{R}^n$. I know that I should use Riesz ...
0
votes
1answer
54 views

The quotient embedding of tensor product

Here is a quotation of a book: Let $A, B$ be two $C^*$-algebras and $J\subset A$ be a $C^*$-subalgebra, then there is a dense embedding $$\frac{A\odot B}{J\odot B}\hookrightarrow\frac{A\otimes_{\...
1
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1answer
42 views

Exact sequence of tensor product

Here is a quotation of a book: Proposition 3.7.1. If $0 \rightarrow J \rightarrow A \rightarrow (A/J)\rightarrow 0$ is an exact sequence, then for every $B$, the natural sequence $$0 \rightarrow ...
1
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1answer
28 views

Easy question about subdifferential of a functional on $L^2(0,T;L^2)$

Define $J:L^1(0,T;L^1) \to \mathbb{R}$ by $$J(v) = \int_0^T \int_\Omega \Psi(v)$$ where $\Psi(v) = \int_0^v \beta(s)\;ds$ where $\beta$ is a nice function that passes through the origin. We have ...
0
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1answer
62 views

Can I apply Lions--Aubin lemma on $X_0 \subset X \subset X_1$ with $X\equiv X_1$?

The lemma is Let $X_0$, $X$ and $X_1$ be three reflexive Banach spaces with $X_0 \subset X \subset X_1$. Suppose $X_0$ is compactly embedded in $X$ and $X$ is continuously embedded in $X_1$. Let ...
1
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0answers
221 views

Adjoint of sum = sum of adjoints

is $\mathcal{D}(A)=\mathcal{D}(B)$ a sufficient condition for $(A+B)^*=A^*+B^*$ , where $A$ and $B$ are densely defined (not necessarily symmetric) operators on some Hilbert space?
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1answer
57 views

Approximation theory and proximinal sets

The question is to give an example such that the finite union of proximinal sets is not proximinal. I have no idea to construct any example to suit this problem, will anybody help me?
2
votes
1answer
81 views

Question about compact operator

So here is my question, Let $H$ be a Hilbert-space and $K:H\rightarrow H$ a compact operator. I know that if $K$ is self adjoint, then it has one eigenvalue $\mu$ such that $|\mu|=||K||$. Can some ...
1
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1answer
44 views

$T$ has a finite rank $\iff$ $\exists N \in \mathbb{N}$ such that $\lambda_n=0$, $\forall n \geq N$

The question goes as follows: $T$ has a finite rank $\iff$ $\exists N \in \mathbb{N}$ such that $\lambda_n=0$, $\forall n \geq N$. Given is the data: $X$ is a Hilbert space with an orthonormal ...
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2answers
97 views

Book on periodic Schrödinger operators

I am looking for good books about the spectral theory of periodic (1-dimensional) Schrödinger operators on a compact interval. A good reference I found was Reed/Simon Analysis of Operators (and a ...
2
votes
2answers
142 views

Fredholm Index: Finite Corank $\Rightarrow$Closed Range [duplicate]

Obviously closed subspaces turn quotient spaces into normed spaces rather than just merely vector spaces. However the dimension involved in Freholm's index are purely algebraic. Why do we thus ...
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2answers
46 views

Can one prove this fact about Fredholm operators like this

If $X,Y,Z$ are Banach spaces and $u: X \to Y. v: Y \to Z$ are Fredholm then $\mathrm{ker}(vu)$ is finite dimensional. Can one argue as follows?: If $x \in \mathrm{ker}(vu)$ then either $x \in \...
1
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1answer
281 views

Counterexample to Marcinkiewicz

I have a version of Marcinkiewicz: Let $(X,\mu)$ and $(Y,\nu)$ be measure spaces and let $1<p_1 \leq \infty$. Suppose that $T$ is a mapping from $L^1(X,\mu) + L^{p_1}(X,\mu)$ to $\mu$- measurable ...
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1answer
61 views

Showing that a certain operator is compact

So here is my problem, I try to show that following operator is compact, \begin{align} J: h_1 & \rightarrow\ell^1 \\ (x_n) & \mapsto(x_n) \end{align} where $$h_1:=\left\{x_n\in \...
4
votes
2answers
95 views

Question about a counterexample concerning compact operators

Does anybody know if the following is true, Let $H$ be an infinite dimensional Hilbert-space and $K:H\rightarrow H$ a compact operator. Then if $|\mathrm{spec}(K)|<\infty$ i.e the spectrum is ...
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0answers
37 views

Characterizing direct sums

Let $U,V$ be vector spaces. Let $T: U \to V$ be a linear map. The codimension of $T$ is defined to be $\mathrm{dim}(V) - \mathrm{dim}(\mathrm{im}(T))$. My questions are: (1) given the subspace $\...
0
votes
1answer
55 views

Question if an operator is compact

So here is my problem, Let $$J_p:\ell^p\rightarrow c_o$$ be the canonical embedding where $c_0:=\{x_n\subseteq\mathbb C:x_n\rightarrow 0\quad n \rightarrow\infty\}$. I have to decide whether the ...
0
votes
1answer
35 views

Pointwise convergence in two variables.

I'm not sure about the following (taken from a proof). If $x \rightarrow x_0$ and $r\rightarrow r_0$ then $\chi_B(r,x) \rightarrow \chi_B(r_0,x_0)$ on $\mathbb{R}^n - S(r_o, x_o)$, where $S(r,x) = \{...
0
votes
1answer
33 views

Weak Boundness in $\mathbb{R}^n$

"Consider $(\mathbb{R}^n,|\cdot|)$ and let $\{e_1,\dots,e_n\}$ be its canonical basis. $B$ is bounded in $\mathbb{R}^n$ iff there exists $M$ such that $$ |(e_i,x)|=|x_i|\le M\qquad\forall x\in B,\...
2
votes
2answers
60 views

Doesn't this theorem hold for general normed spaces

My question is: does this hold for any normed space $X$ or only for Banach spaces: If $X$ is a Banach space then $K(X)$ (space of compact operators) equals $B(X)$ (space of bounded operators) if ...
3
votes
1answer
139 views

Similarity transformation of a linear operator

I've seen in some books that given a differential operator $$\frac{d}{dx}$$ under a similarity transformation we get $$\frac{d}{dx}\rightarrow T\frac{d}{dx}T^{-1}=\left(\frac{d}{dx}-\frac{\dot{T}}{T}...
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2answers
38 views

Inquiry about operator algebra

I've just began studying some quantum mechanics, and I'm not sure why certain rules in operator algebra are correct. For instance, in this book it is stated that $$\left(\frac{d}{dx}+v(x)\right)\left(...
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0answers
31 views

Question concerning continuity of some linear map

So here is my question, I wanted to prove that the canonical embedding of $\ell^p(\mathbb N,\mathbb C)$ in $c_0:=\{(x_n)_{n\in\mathbb N}\subseteq\mathbb C:\lim_{n\rightarrow\infty}x_n=0\}$ is ...
1
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1answer
53 views

Show that a function is bounded.

Let $ f $ be a bounded continuous function on $ \mathbb R^3$ that supported on the unit ball $ B(0,1)$ and satisfies the condition $ \sup_{ x, y \in \mathbb R^3, x \neq y } \frac{ |f(x) - f(y) | }{ |x ...
5
votes
1answer
165 views

Sobolev spaces in one-dimensional vs multidimensional

Here in Wikipedia, it is said that in the one-dimensional case, it is enough to assume that the $(k-1)$-th derivative of the function $f$, is differentiable almost everywhere and is equal almost ...
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0answers
48 views

Convergence of product of continuous functions and test functions

I suspect the following result is true but I"m not sure how to go about proving: It is given that $\Omega \subset \mathbb{R}^{n}$ is an open bounded, connected domain.(Not sure if theses conditions ...
6
votes
1answer
545 views

Connection between separable measure spaces and $\sigma$-finite measure spaces

I recently came across a theorem which makes a hypothesis that a certain measure space is separable (the definition can be found here). In order to avoid confusion, I'll add the definition here: We ...
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1answer
32 views

A simple question about Lance's weak expectation property.

Here is a quotation of a book: Definition 3.6.7. A C*-algebra $A\subset B(H)$ is said to have Lance's weak expectation property (WEP) if there exists a u.c.p map $\Phi: B(H)\rightarrow A^{**}$ ...
3
votes
1answer
62 views

Question about an integral operator

So here is my question, I know that the operator $$T:L^2[0,1]\rightarrow L^2[0,1]$$ $$f\mapsto(Kf)(x)=\int_{[0,1]}k(x,y)f(y)\;dy$$ for a function $k$ continuous on $[0,1]^2$ is compact. Is this also ...
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1answer
36 views

A equivalent proposition of contractive completely positive map

Proposition 3.6.6. Let $A\subset B$ (C*-algebras) be an inclusion. Then the following are equivalent: (1). there exists a c.c.p.(contractive completely positive) map $\phi: B\rightarrow A^{**}$ such ...
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1answer
53 views

A proof of a proposition of tensor product

Proposition 3.6.5.(The Trick) Let $A\subset B$ and $C$ be C*-algebras, $||.||_{\alpha}$ be a C*-norm on $B\odot C$ and $||.||_{\beta}$ be the C*-norm on $A\odot C$ obtained by restricting $||.||_{\...
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1answer
672 views

convolution of function with itself 4 times

I have to compute the convolution of $ f(t) = \frac{1}{\pi}\frac{1}{t^2 + 1} $ with itself 4 times, i.e. $$ f \star f \star f \star f $$ I slightly doubt that doing it in steps, i.e. taking $f \star ...
0
votes
1answer
127 views

Proof using closed graph theorem

Let $X$ be a Banach space. Let $A:X\rightarrow X$ be a linear map such that $\forall \phi \in X' : \phi \circ A$ is continuous. Prove that $A$ is continuous. How can I prove it using Closed Graph ...
2
votes
1answer
75 views

Please verify this definition of locally convex vector space

A locally convex vector space is a vector space $V$ equipped with a family $P$ of separating semi-norms. Is this a correct definiton? So to determine whether a norm induces a locally convex topology ...
0
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1answer
60 views

Some infinite dimensional linear algebra, kernels of linear maps

I'm studying functional analysis (namely weak convergence) and need to prove the following result: if $f,f_1,\ldots f_n$ are some linear maps $X\to \mathbb{C}$, where $X$ is a vector space over $\...
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1answer
75 views

Separating family of seminorms

Let $(p_n)_{n=1}^{\infty}$ be a family of seminorms on a vector space $X$. Assume that series $\sum_{n=1}^{\infty} p_n(x)$ is convergent for any $x \in X$ and let's denote sum of such series as $q(x)$....
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2answers
50 views

Identity in inner product space

Let $x$ and $y$ be elements of inner product space, such that $\| x+2y \|^2=5$ and $\| 2x+y \|^2=4$ Prove that $9 \| x+y\|^2 + \| x-y\|^2 =18$ Any advice?
2
votes
1answer
133 views

Locally convex topological vector space

In $C(\mathbb{R})$ space we define two families of seminorms: $p_x(f)=|f(x)|$ where $ x\in\mathbb{Q}$ and $q_x(f)=|f(e^x)|$ where $ x \in \mathbb{R}$ I have to check if above families induce locally ...
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0answers
74 views

A question on functional analysis

Let $H_i$, where $i = 1,2$ be Hilbert spaces and $T_i : H_i \rightarrow H_i$ be closed operators, such that $T_i$ have positive spectrum. Let $\phi : H_1 \rightarrow H_2$ is an isometric isomorphism ...
1
vote
1answer
53 views

Change of variable in Hardy Littlewood proof

This is part of a proof I try to understand. Lets $Tf(x)$ be the Hardy littlewood maximal funtion, $$Tf(x) = \sup_{r>0} \frac{1}{B(r,x)} \int_{B(r,x)} f(y) dy$$ and $E_\lambda = \{y: |Tf(y) |> ...