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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74 views

Comparing weak and weak operator topology

We can compare topologies on $B(H)$. For instance, Sot topology is stronger than wot topology or $\sigma-$ weak topology is equivalent to weak* topology. I would like to compare wot topology and weak ...
3
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1answer
72 views

A question about local convexity of the weak operator topology

By definition, I know a locally convex space is a topological vector space whose topology is defined by a family of seminorms $\cal P$ such that $$\bigcap_{p\in{\cal P}}\{x\colon p(x)=0\}=\{0\}.$$ ...
2
votes
1answer
61 views

Prove there cannot be an inner product which turns $l^p$ into an inner product space?

For all $1\leq p < \infty, \mbox{ }p$ is not equal to 2, prove there cannot exist an inner product that turns $(X,\|\cdot \|_p)$ into an inner product space; that is, prove that there cannot be ...
2
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1answer
73 views

Closed Operators: Spectrum

Given a Hilbert space $\mathcal{H}$. Consider operators: $$T:\mathcal{D}(T)\to\mathcal{H}$$ Suppose one has: $$T=\overline{T}=T^{**}$$ Then it may happen: $$\sigma(T)=\varnothing,\mathbb{C}$$ What ...
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2answers
113 views

about weak convergence in $L^2(0,T;H)$.

Exercise Suppose $H$ is Hilbert space and $u_k$ converges weakly to $u$ in $L^2(0,T;H)$. Suppose further we have the uniform bounds $\mathrm{esssup}_{0≤t≤T} ||u_k(t)||≤C$. Then ...
0
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1answer
63 views

Eigenvalue of Compact Operators

To prove that the set of eigenvectors of a compact linear operator on a normed space $X$ is countable, I read "it suffices to show that for every real $k > 0$ the set of all eigenvalues whose ...
1
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1answer
72 views

A Question on Compact Operators on Hilbert Space

I read this question which I have no idea how to start. Could anyone provide me with some detailed answer, please? Thanks. Suppose that a linear operator $F$ from a Hilbert space $\mathcal H$ to ...
5
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2answers
87 views

Why $\rho(AABABB)=\rho(ABAABB)$?

Let $A,B$ be two matrices, $\rho$ be spectral radius, which is the top eigenvalue of a matrix. I discovered that $$\rho(AABABB)=\rho(ABAABB).$$ But I could not find the reason. By the way, all I had ...
1
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1answer
113 views

Inner products of weakly convergent sequences

I have a weakly convergent sequence in $L^2(U)$ (for $U$ some bounded open domain with smooth boundary), $u_k\rightharpoonup u$. I want to show that there is a sequence $v_k\rightharpoonup v$, such ...
0
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1answer
25 views

Is dimension unique?

To what extent can we describe a space as being uniquely n-dimensional? For example, the space of functions on I=[0,1] are frequently described as infinite dimensional, where I serves as the index for ...
0
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2answers
47 views

Show existence and uniqueness of integral equality with neumann-series

I want to show that for $$x(s)-\int_0^12rs\cdot x(r)dr=\sin(\pi s)$$ there exists exactly one solution $x \in C^0([0,1],\mathbb R)$.
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0answers
25 views

Something like a trace inequality of $H^1(\Omega)$

I have the following question: Let $\Sigma$ an a surface inside of an open domain $\Omega\subseteq\mathbb{R}^3$, where $\Sigma$ divides $\Omega$ in 2 open domains (for example: $\Sigma$ could be a ...
2
votes
1answer
253 views

Positive operator has a positive spectrum?

Let $T : \operatorname{dom}(T) \rightarrow H $ be a positive self-adjoint operator, is it then true that $\sigma(T) \subset [0,\infty)$? This is something that sounds natural and I guess that it is ...
0
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2answers
68 views

Spectral Measures: Numerical Range

Given a Hilbert space $\mathcal{H}$. Consider a normal operator $N:\mathcal{D}(N)\to\mathcal{H}$. The goal here is to prove: $$\langle\sigma(N)\rangle=\mathcal{W}(N)$$ By a previous result one has: ...
0
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1answer
67 views

An inequality about operator norm

Let $H$ be a Hilbert space and $T\in B(H)$, with $T_{i}\rightarrow T$ in strong operator topology. Then can we prove that $\liminf_{i\rightarrow \infty}||T_{i}||\geq ||T||$ ?
2
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1answer
75 views

A question about maximal and minimal tensor product

Let $A, B$ be two C*-algebras and $\pi: A\otimes_{\max} B\rightarrow M_{n}(\mathbb{C})$ be a representations, then this $\pi$ can factor through the minimal tensor product $A\otimes_{\min} B$ ? (That ...
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1answer
157 views

Real Analysis vs. Functional Analysis version of Arzela-Ascoli Theorem

Consider a collection of functions defined on a compact set such that they are uniformly bounded and equicontinuous. Then 1) Real analysis version: Every sequence in that collection will contain a ...
14
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1answer
215 views

What's the minimal structure needed to define a notion of derivative?

I know that, for example, to define a limit all you need is the notion of "closeness" generated by a topology; and to define an integral you need a measure function and a sigma-algebra on which it is ...
2
votes
1answer
116 views

Segment ordered density conjecture.

I have a set $S\subset\mathbb {R}^2$ with the following property (P) $\forall x,y\in S$, $\forall\mathscr{C}$ a convex set that contains $x$ in its interior, $bd\mathscr{C}\cap [x,y]\subset ...
2
votes
2answers
82 views

Relation between $f(x+y)$, $f(x)$ and $f(y)$

Given a function $f:\mathbb{R} \to \mathbb{R}$ such that $$f(x)=x+\int_{0}^{x}f(t)\,dt$$ then what is the relation between $f(x+y)$, $f(x)$ and $f(y)$ My Try: we have ...
5
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0answers
229 views

Is the left regular representation of an algebra, always faithful?

Let $\mathcal{A}$ be a unital associative algebra with a countable basis $\mathcal{b}$ over $\mathbb{C}$. Let $H=l^2(b)$ be the Hilbert space generated by $\mathcal{b}$. Let $H_0 = \{v \in H \ \vert \ ...
2
votes
1answer
98 views

Weak convergence in $\mathcal{l}_p$ and coordinatewise convergence

Let $x^n=(x^n_1, x^n_2,...)$ be a bounded sequence in $\mathcal{l}_p$ for $1<p<\infty$ and such that $x^n_i$ converges to $x_i$ for all $i\in\mathbb{N}$. I'm trying to prove that ...
1
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0answers
41 views

Does an inequality between kernels imply an inequality between the norms of integral operators?

Assume that $g(x,y)$ and $h(x,y)$ are two positive functions such that $0<g<h$ and assume that $$T_g, T_h : L^2(B^n,R)\to L^2(B^n,R)$$ are integral operators defined by $$T_k[f](x)=\int_{B^n} ...
1
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1answer
153 views

Almost everywhere convergence of a bounded sequence in $H_0^1(\Omega)$

Let $\Omega \subset \mathbb{R}^N$ be a smooth domain with bounded complement (e.g. $\Omega = \mathbb{R}^N \setminus \overline{B(0;1)}$), and $(u_n)$ be a bounded sequence in $H^1_0(\Omega) = ...
6
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1answer
152 views

An argument from a blog article of Terence Tao

Let $A_1, A_2, A_3, \ldots , A_m$ be positive semi-definite Hermitian matrices and then consider the polynomial $p(z,z_1,z_2,\ldots,z_m) = \det(z+z_1A_1 + z_2A_2 + \cdots+z_mA_m)$ Now Tao argues that ...
2
votes
2answers
80 views

Positive elements of a $C^*$-algbera form a poset

My knowledge of $C^*$-algebras is very little. We call an element positive if $a=b^*b$ for some $b$ and make a relation on all positive elements by saying $$ b \geqslant a \iff b-a \text{ is ...
1
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1answer
156 views

Show that operator T is a contraction mapping

I want to check whether the operator T defined as: $Tf(x) = \beta \left\lbrace \sum_{\theta} \mu_\theta \left[ h_\theta(x) + \int f(x') Q_\theta(x,dx') \right]^\alpha \right\rbrace^{1/\alpha} $ is a ...
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0answers
164 views

Implicit function theorem to prove tangent plane to the surface

Let $\Phi$ be the regular surface at $(u_o,v_o)$ (ie., $\Phi$ is of class $C^1$ and $T_u\times T_v\ne 0$ a)Use the implicit function theorem to show that the image of $\Phi$ near $(u_o, v_o)$ is the ...
2
votes
1answer
170 views

Classification of operators

I have a collection of questions about the limit point/circle concept and self-adjointness that are kind of connected, so I would like to ask them in a row. Apparently, an operator that is limit ...
1
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1answer
79 views

Confusion about the definition of self adjoint and formally self-adjoint

I have some confusion about the definition of self-adjoint operators and formally self-adjoint operators. Let me write down the background information. Let $H$ be a infinite dimensional complex ...
0
votes
1answer
89 views

Is the following true or false?

I need to prove or counterexample for: If $f\in L^1 ([-\pi,\pi])$ and $\phi(n)$ be an orthonormal sequence and $(f,\phi(n))=0$ for all integers $n$ then $f=0$ a.e.
5
votes
1answer
226 views

Can the composite of two projections really fail to be a projection?

Let $H$ denote a Hilbert space. For any closed subspace $C \subseteq H$, write $P_C$ for the orthogonal projection onto $C$. Then according to wikipedia, the composite $P_U \circ P_V$ needn't be a ...
1
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1answer
53 views

I need help with a proof: invertibility of $b-\lambda$ in $B$ iff $b-\lambda $ invertible in $A$

Let $A$ be a unital $C^\ast$ algebra and let $B$ be a $\ast$ subalgebra such that $B \oplus \mathbb C = A$ and such that the unit in $B$, $1_B$, is not equal to the unit in $A$. I am trying to show: ...
0
votes
1answer
46 views

spectrum of compact operators

Let $\phi\in\ell^\infty(\mathbb{N})$. For $p\in[1,\infty]$, define $$M_\phi:\ell^p\to\ell^p,\quad f\mapsto\phi f.$$ Use spectral theory to show that, if $M_\phi$ is compact, then $\phi\in c_0$. Here ...
5
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1answer
51 views

multiplicative semi-norms on $\mathbb{C}[x]$

A multiplicative semi-norm on a ring $A$ is a function $|\,|:A\to \mathbb{R}_{\ge 0}$ that is multiplicative and satisfies the semi-norm conditions: $|0|=0,|1|=1\\ |fg|=|f||g|,\\ |f+g|\le |f|+|g|.$ ...
3
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0answers
24 views

Inverse continuous

According Dehman, Gérard and Lebeau in "Stabilization and control for the nonlinear Schrödinger equation on a compact surface" (Math. Z. (2006) 254:729–749, DOI 10.1007/s00209-006-0005-3) is claimed ...
0
votes
1answer
41 views

How to restore a function from its Fourier transform on the imaginary axis?

Let $f$ be a `very good' function on the real line; say, infinitely differentiable and compactly supported. We are given its Fourier transform on the imaginary axis: $$g(x)=\int_{\mathbb ...
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6answers
435 views

Possible flaw in “proof” that a sum of two compact operators is compact

If X and Y are Banach spaces, and $A: X \to Y$, $B: X \to Y$ are both compact operators, then $A + B$ is compact. A + B is compact if and only if for every bounded sequence $\lbrace x_n \rbrace$ ...
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2answers
87 views

equivalent metric space

Let $(X, d)$ be a metric space where $d$ is unbounded, that is, $$\sup\{d(x; y) : x, y\in X\} = \infty$$ Define a bounded metric $p$ on $X$ such that: $(i).$ $f : (X, d) \rightarrow (X, p)$, $f(x) = ...
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1answer
144 views

How to read an expression with sup and inf

Let $K:H \to H$ be a linear operator on a Hilbert space $H$. The action of $K$ on any $u$ is denoted by $Ku$. I define the domain of $K$ as the set $D(K)$ of all $u \in H$ with the property that ...
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2answers
36 views

Finding multiple functions with same $f_{even}$ but different $f_{odd}$?

A function can be decomposed as $f(x) = f_{even}(x) + f_{odd}(x)$ where $f_{even}(x)=\dfrac{f(x)+f(-x)}{2}$ and $f_{odd}(x)=\dfrac{f(x)-f(-x)}{2}$. If we know only $f_{even}$, how can we find ...
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1answer
20 views

When can we say that $T\in B(H)$ is $YS$ for some fixed $S$ in $B(H)$?

Let $T,S\in B(H)$ where $H$ is a Hilbert space. Suppose that for all $x\in H$, $\|Tx\|\leq \|Sx\|$. Could we then say that $T=YS$ for some $Y\in B(H)$? Would it help if $S$ were a contraction or ...
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1answer
55 views

Example of a subspace S of a Hilbert space such that S^(⊥⊥) does not equal S?

I try to find an example of a subspace S of a Hilbert space H such that S^(⊥⊥) does not equal S. I know that subspace cannot be closed as for closed subspaces S^(⊥⊥)=S holds true. Does there exist ...
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2answers
52 views

A question about inclusion of $L^r(\mu)$ spaces for different $r$ and different measures $\mu$

For some measures, the relation $r<s$ implies $L^r(\mu)\subseteq L^s(\mu)$ ; for others, the inclusion is reversed; and there are some for which $L^r(\mu)$ does not contain $L^s(\mu)$ if $r\ne ...
3
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1answer
74 views

(Possible) application of Sarason interpolation theorem

This question is related to the following Wikipedia article on Nevanlinna–Pick interpolation. At the end it has been written as Pick–Nevanlinna interpolation was introduced into robust control by ...
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1answer
63 views

Does Hilbert–Schmidt theorem imply the space is separable?

The Hilbert–Schmidt theorem says a self-adjoint compact operator on a Hilbert space have a complete orthonormal set consisting of eigenvectors. Does that imply the space is separable?
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1answer
64 views

Need some help understanding one step in this proof of homeomorphism $\Omega (C(X)) \cong X$

Let $X$ be a compact Hausdorff space and $\Omega (C(X))$ the space of characters on $C(X)$. I am showing that the map $x \mapsto e_x$ where $e_x$ is evaluation at $x$ is surjective (I already showed ...
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0answers
44 views

Compact operators on a Banach space $X$ are closed in the bounded operators on $X$. - Proof correction help

I am given a proof of the following statement (see below). Compact operators on a Banach space $X$ are closed in the bounded operators on $X$. I was told that there is an error in this proof - I ...
0
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2answers
41 views

$T_1,T_2,S$ bounded linear operators on a Banach space $X$, with $T_1, T_2$ compact. Show that $T_1+T_2$, $\alpha T_1$, $T_1S$, $ST_1$ are compact.

Here is my question: Let $X$ be a Banach space and suppose $T_1$, $T_2$, $S$ are bounded linear operators from $X$ to $X$, with $T_1$ and $T_2$ compact. Show that $T_1+T_2$, $\alpha T_1$, $ST_1$ and ...
1
vote
1answer
84 views

finding a counter example to Caratheodory's convex hull theorem for infinite dimentional space

Recently I learned in class Caratheodory's convex hull theorem which suggests that in a finite dimentional (n) space every x can be written as a convex-combination of maximum (n+1) vectors. I was ...