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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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 ...
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1answer
34 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 ...
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47 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 ...
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1answer
29 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 ...
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1answer
24 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 ...
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1answer
40 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 ...
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80 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
27 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?
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1answer
58 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 ...
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1answer
32 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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66 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 ...
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2answers
52 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
44 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 ...
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1answer
62 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
50 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 ...
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2answers
73 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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33 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 ...
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1answer
44 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 ...
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1answer
30 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) = ...
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1answer
27 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 ...
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2answers
55 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 ...
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49 views

does the limit of the ratio of $p+1$ norm and $p$ norm equal to $\infty$ norm [duplicate]

Suppose that $f\in L^1(\Omega,\Sigma,\mu)\cap L^\infty(\Omega,\Sigma,\mu)$. Then I have proved that for any $1\leq p\leq \infty$, $f\in L^p(\Omega,\Sigma,\mu)$. Moreover, I have proved ...
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1answer
65 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 ...
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2answers
32 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 ...
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30 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 ...
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1answer
47 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 ...
4
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1answer
70 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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40 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 ...
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214 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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36 views

The canonical quotient map between two tensor product [duplicate]

Let $A, C$ be two C*-algebras. Does there exist a canonical quotient map from $A\otimes_{max} C\rightarrow A\otimes C$? $A\otimes_{max} C$ (resp. $A\otimes C$) denote the completion of $A\odot B$ ...
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1answer
17 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 $\Phi: B(H)\rightarrow A^{**}$ ...
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1answer
44 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
27 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
31 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
140 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 ...
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1answer
53 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
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1answer
47 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 ...
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1answer
45 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
45 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 ...
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2answers
41 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?
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1answer
89 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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65 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 ...
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1answer
27 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) |> ...
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1answer
103 views

What makes compact operators special?

I would like to understand why compact operators are considered so special to consider them as an extra class of operators. Over Hilbert spaces these (as far as I know) these are the ones with ...
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0answers
19 views

Minimizing functionals with constraints

The Lagrange multiplier method allows one to minimize a multi-variable function with respect to constraints; is there an analogous or similar procedure for functionals of many functions with ...
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2answers
98 views

Examples of skew adjoint differential operators

I just need some references which studies examples of skew adjoint differential operators generating unitary strongly continuous groups of operators, and its applications to partial differential ...
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1answer
95 views

Compact Operator <=> Separable Range

Is it true that a bounded operator is compact iff its range is separable: $$T\text{ bounded}:\quad T\text{ compact}\iff \mathcal{R}(T)\text{ separable}$$
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1answer
40 views

Existence of nontrivial bounded linear operator?

Are there two normed spaces such that there is no nontrivial bounded linear operator between them: $$\nexists T:X\to Y: T\text{ nontrivial, linear and bounded}$$
2
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1answer
56 views

Completion, injection

$\mathcal{H}$: Real hilbert space with inner product $(\cdot,\cdot)$. $D$ is a subspace of $\mathcal{H}$. Let $\mathcal{E}$ with domain $D$ be a positive definite (i.e. for all $u \in D $, ...
2
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1answer
50 views

Operator, closed

We consider the concrete Hilbert space $L^{2}(E,m)=L^{2}(E,\mathcal{B},m)$ with usual inner inner product $(\cdot,\cdot)$ where $(E,\mathcal{B},m)$ is a measure space. For $u,v:E\to \mathbb{R}$,we ...