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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3
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56 views

duality of $L^p$ spaces with non $\sigma$ finite measure

Why is the condition that $\mu$ (measure) is $\sigma$-finite is important for the relation $(L^1)^{*} = L^{\infty}$? This condition is added while proving that $(L^p)^{*} = L^{q}$ where $p, q$ are ...
2
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0answers
48 views

A sequence of strongly continuous one-parameter unitary groups

Suppose that for a sequence $\{A_n\}_n$ of bounded self-adjoint operators in a Hilbert space $\mathcal H$ we have $e^{itA_n} \to e^{itA}$ strongly, for all $t \in \mathbb R$, where $A$ is a (possibly ...
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1answer
77 views

Weak derivative (Sobolev spaces)

I'm reading "Functional Analysis" - Michel Willem and I can't understand the definition of weak derivative from chapter 6, namely the definition of $$ \partial^\alpha f. $$ Can you give me a concrete ...
0
votes
2answers
100 views

question about the proof that the set C[a,b] with uniform norm is complete

I am trying to understand the proof that the set of continuous function is complete under uniform/supremum norm. First, suppose we have a Cauchy sequence of continuous functions ${f_n(t)}$ with ...
1
vote
1answer
54 views

is $L^2 (\mathbb R)\subset L^\infty(\mathbb R)$?

I know that because i'm working on an infinite-measure space it could be tricky. And from my experience the answer to my question is probably no.. But nevertheless, I can't think of a non-bounded ...
4
votes
1answer
41 views

Seminorms in distribution theory are norms, right?

In distribution theory the seminorms that you use there $p_m( \phi) := \max_{|\alpha| \le m} \sup_{x \in \Omega} |(\partial^{\alpha}(\phi) (x)|, \phi \in C_c^{\infty}(\Omega)$. Those guys are norms ...
3
votes
1answer
58 views

Gaussian, measurability

I have a quesition about an isonormal Gaussian process and measurability. Let $\mathcal{H}$ be a real separable Hilbert space with inner product $\langle \cdot,\cdot \rangle$ and norm ...
5
votes
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85 views

What does a well-posed problem imply?

A well-posed problem in the sense of Hadamard states that: A solution exists The solution is unique The solution's behavior changes continuously with the initial conditions. Now in order to prove ...
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0answers
28 views

Is there a proper subspace of $T$ that includes $T^{n}x$ for all $n\in \mathbb{N}$?

Suppose $E$ is a normed space, $T$ is a bounded operator from $E$ to $E$ and $B_E$ is closed unit ball of $E$. If there is $\exists \epsilon >0$ and $% \exists y\in B_{E}$ such that $\left\Vert ...
1
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2answers
23 views

Proving a property for a metric space

Let $(X,d)$ be a metric space. And it also has the property $d(x_1+x_2,y_1+y_2)\leq d(x_1,y_1)+d(x_2,y_2).$ Is it also true that $d(x_1+x_2+...+x_n,y_1+y_2+...+y_n)\leq ...
3
votes
1answer
96 views

Fredholm Alternative and Compact operator

I m working on the following problem: Let $K$ be a compact operator on a Hilbert space, $H$, and let $K^*$ be its adjoint. For each $\lambda \in \mathbb{C}$, define $$N_\lambda=N(\lambda I-K), ...
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1answer
66 views

Question about the relation between the adjoin and inverse of linear operator on Hilbert space

I am teaching myself functional analysis from a CS background. I am clueless about the following exercise problem of introductory functional analysis. Any hint or help is appreciated. Thanks a lot! ...
3
votes
0answers
56 views

Relating Fourier transform theory on two distinct subspaces

In Fourier transform theory (on $\mathbb{R}$), three vector spaces play a very important role: $L^1(\Bbb R)$, $L^2(\Bbb R)$ and the Schwartz space $\mathcal{S}(\Bbb R)$. Arguably the nicer spaces of ...
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1answer
44 views

Space of Continuous functions from Hilbert Cube is separable

I am trying to prove that $C([0,1]^\mathbb{N})$, the space of continuous functions from $[0,1]^\mathbb{N}$ into a scalar field, is separable. I am able to prove that $C([0,1])$ is separable using ...
1
vote
1answer
102 views

how to prove this epsilon-delta property for continuous functional calculus with normal elements?

Let $ A$ be a C* algebra, $f\in C([-1,1])$. Prove that for every $\epsilon >0, \exists \delta >0,$ s.t. for $\forall x \in A, x=x^*, \| x \| \leq 1$ and $\forall y \in A, \|y\| \leq 1$, we have ...
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1answer
45 views

A Boundedness and Precompactness question

Let $l_{1+} = \{ x=(x_n)_{n=1}^{\infty} : p_{\alpha}(x):=(\sum_1^{\infty} |x_n|^{\alpha})^{1/\alpha} < \infty, \forall \alpha >1 \}$, and let the topology be given by the norms ...
2
votes
1answer
42 views

Projection-valued measure property

I am reading the book functional analysis from Dirk Werner. In this book he introduces a Projection-valued measure (PVM) as follows. Let $L(H)$ denote the set of linear,bounded operators $T:H\to ...
3
votes
1answer
81 views

How to characterize the boundary of a convex set?

I am working on a part of a paper related to topological properties of boundary points. It is important to me to realize the topological and algebraic behavior the boundary points of convex sets. I ...
0
votes
2answers
65 views

inequality regarding norm of linear operator

Let $T$ be a linear operator on a vector space $X$. For $x\in X$, I know there's the inequality that $$||Tx||<||T||||x||$$ Yet I'm wondering what are those norms. Are they arbitrary? especially on ...
1
vote
2answers
76 views

Compact operators on Hilbert Space

I m working on the following problem: Let $K:H\rightarrow H$ be a compact operator on a Hilbert space. Show that if there exists a sequence $(u_n)_n\in H$ such that $K(u_n)$ is orthonormal, then ...
6
votes
1answer
226 views

Converse of uniform boundedness principle

The uniform boundedness principle says if we have a collection of bounded linear operators $\Gamma$ from a banach space $X$ into a normed vector space $Y$, which is pointwise bounded on $X$, i.e. ...
1
vote
1answer
98 views

Sobolev spaces and Fourier transform

Does the equation $$\partial^{\alpha} (Ff)(y) = (-i)^{|\alpha|} F(x^{\alpha} f)$$ still hold for $|\alpha| \le m$ and $f \in H^m(\mathbb{R}^n) = W^{m,2}(\mathbb{R}^n)$$? $F$ denotes the Fourier ...
2
votes
0answers
52 views

Nonlinear boundary value problem with Newton's Iteration

Consider the nonlinear two-point boundary value problem: $-u''(t)+u(t)^p=\phi(t)$ ($0\leq t\leq 1$, $p\geq 2$ and $\phi\in C[0,1]$ is given function) $u(0)=u(1)=0$ Given the following ...
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0answers
66 views

Examples for when differentiability fails

Let $l^1(\mathbb{N};\mathbb{R})$ be the set of all sequences $\mathbb{N}\to\mathbb{R}$ such that $\sum_{n\in\mathbb{N}}|x_n|<\infty$ for all $x\in l^1(\mathbb{N};\mathbb{R})$, together with the ...
0
votes
1answer
33 views

In the definition of the strong operator topology on $L(\mathcal{X}, \mathcal{Y})$, why must $\mathcal{X}$ and $\mathcal{Y}$ be Banach spaces?

In Folland, the strong operator topology is defined as the topology on $L(\mathcal{X}, \mathcal{Y})$ induced by the evaluation maps $\{T \mapsto Tx\}_{x \in \mathcal{X}}$, where $\mathcal{X}, ...
0
votes
1answer
61 views

Resolvent: Decay Behavior

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote resolvent set: ...
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0answers
57 views

Every Hilbert space is connected

Let $H$ be a Hilbert space. Proving $H$ is connected, suppose $\{e_i\}_{i\in I}$ is a orthogonal basis of $H$. Thus $H=\bigoplus_{i\in I} \Bbb C e_i$. Clearly $\Bbb Ce_i$ is connected for every $i$, ...
2
votes
2answers
189 views

Tent map invariant density

Is there a formula for the invariant density for the tent map $f_t$ (for $\sqrt{2}\leq t\leq 2$)? $$ f_t = \begin{cases} tx, & 0\leq x<1/2 \\ t-tx, & 1/2 \leq x\leq 1. \end{cases} $$ By ...
0
votes
1answer
16 views

Traingular inequality for $L_\infty$ norm

Consider $f$ which is a measurable function on $X$. We define $\|f\|_{\infty} = \inf \ \{a \geq 0: \mu(\{x:|f(x)| > a \}) = 0\}$ with the convention that $\inf \phi = 0$. Define $L^{\infty} = \{ f: ...
-2
votes
1answer
35 views

Proving the weak topology coincides with the original topology [closed]

Let $F$ be a family of real valued continuous functions on topological space $(X,J)$. If for each closed set $C$ in $(X,J)$, and each $x\notin C$, there exists $f\in F$ such that $f(x)=1$ and ...
1
vote
1answer
243 views

Weak convergence + compactness = strong convergence? [duplicate]

Let $X$ be a Banach space and $K$ a compact subset of $X$. If $(x_n)_n$ is a sequence such that $x_n\in K$ for all $n$ and $(x_n)_n$ converges weakly to some $x\in X$, i.e. $x^*(x_n)\to x^*(x)$ for ...
1
vote
1answer
98 views

properties of a Köthe space s

Could you please help me answering the following question? Consider the Köthe space $K_ {\infty}(n^p) = \{ x= (x_n)_1^{\infty}: |x|_p := \sup_n|x_n|n^p<\infty, \forall p \in \mathbb{N} \}$ with ...
4
votes
1answer
46 views

For any sequence from Frechet spaces there exists a sequence that takes it to zero

I am trying to prove following for Frechet spaces($X$): Show that any sequence $(x_n) \subset X$ there exists a sequence $(\lambda_n)$ with $\lambda_n \neq 0$, $\lambda_n \downarrow 0$ such that ...
1
vote
1answer
45 views

Bijection between a Hilbert and a Banach space

I am working on a question where I have to show that for a Hilbert space $\mathscr{H}$, and closed linear subspace $Y \subset \mathscr{H}$, $\mathscr{H}/Y$ is a Hilbert space, isomorphic to ...
1
vote
1answer
28 views

Does $E(\frac{f(x)}{g(x)})\leq \frac{E(f(x))}{E(g(x))}$ hold?

We current have the relation that $h(x)\leq\frac{f(x)}{g(x)}, f(x)>0, g(x)>0$. Then, can we get that $E(h(x))\leq E(\frac{f(x)}{g(x)})$? Furthermore, does $E(\frac{f(x)}{g(x)})\leq ...
1
vote
1answer
32 views

Bound on the set of compactly supported distributions with support in the same compact set

Consider the set of all compactly supported distributions $v\in\mathcal{\mathcal{E}^{\prime}}(\mathbb{R}^{n})=\left(C^{\infty}\right)^{*}$ with compact support in a fixed compact set $\Omega$ . ...
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vote
0answers
13 views

questions on non-metrizable weak convergence

Let $X$ be a normed vector space. I wonder if it is true If $G\in X^{**}$ such that if $f_n\rightarrow f$ $weak^*$ then $G(f_n)\rightarrow G(f)$. Then $G$ is $weak^*$ continous? thanks for any ...
2
votes
1answer
136 views

weak convergence star $wk^*$

I'm starting to study the weak star ( $wk^*$) topology and I want to solve the following task: Let $X$ banach space and $F\in X^{**}$ (bidual space) such that $Ker(F)$ is $wk^*$ closed then $F$ is ...
3
votes
2answers
76 views

Existence of central cover for a representation of a C*-algebra

I've been trying to learn the basics about the representation theory of C*-algebras and came across the following in Pedersen's C*-algebras and their Automorphism Groups: With each ...
0
votes
1answer
42 views

Functional analysis, normed space

Can you please help me with this exercise.. I have to check if in interval $[a,b]$ continuously differentiable functions $x=x(t)$ norm can be defined as: $$\vert x(b) - x(a) \vert + \max_{a \leq t ...
4
votes
3answers
164 views

Can one find a stronger norm on a Banach space?

Given a Banach space $V$ of infinite dimension with norm $\|\cdot\|_1$, is that possible to find a norm $\|\cdot\|_2$ on $V$ such that the topology induced by $\|\cdot\|_2$ is strictly stronger than ...
0
votes
1answer
36 views

Find a sequence of complex polynomials with certain properties. (Hardy spaces over unit circle)

Let $\lambda\in \Bbb S^1$. Find a sequence of complex polynomials $p_n(z)$ such that for any $c>0$ the following inequality does not hold: $$|p_n(\lambda)|\le c\cdot \|p_n\|$$ where ...
2
votes
1answer
65 views

Spectral Measures: Core Lemma

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a dense domain: ...
2
votes
1answer
129 views

kronecker product of integral operator

I'm wondering whether can we define an explicit Kronecker product of Hilbert–Schmidt integral operators. $T(f)=\int k(x,y)f(y)dy$. Since integral operator is an extension of matrix operation, I'm ...
0
votes
1answer
26 views

Question about ortogonality on $L^2(\Omega)$

Let $u\in L^2(\Omega)$. Is the following proposition true? $\big(\forall v\in H^1_0(\Omega)\big)\quad (u,v)_{0,\Omega}:=\displaystyle\int_\Omega uv=0$ then $u=0$ ? where $H_0^1(\Omega)$ are the ...
2
votes
0answers
89 views

A property of the Baire $\sigma$-algebra

This problem was assigned in a real analysis class I'm in and I have not been able to solve it. Let $X$ be a topological space. For each $x\in X$ let $\varphi_x\colon C(X,[0,1]) \to [0,1]$ be given ...
0
votes
1answer
16 views

Characterization of product of dual dual Hilbert spaces.

Let $X_i$ be Hilbert spaces and $X_i'$ its dual spaces, with $i=1,2$. Let $F\in(X_1\times X_2)'$. Prove that exists $F_1\in X_1'$ and $F_2\in X_2'$ such that ...
0
votes
2answers
113 views

Spectral Measures: Scale Embeddings

Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}N\to\mathcal{H}:\quad N^*N=NN^*$$ And its spectral measure: ...
1
vote
1answer
207 views

Convergence of the spectrum under norm resolvent convergence

Suppose $\{A_n\}$ is a sequence of self-adjoint operators in a Hilbert space $\mathcal H$, and $A$ is a self-adjoint operator, with $A_n \to A$ in norm resolvent sense. Since $A_n \to A$ in strong ...
2
votes
1answer
26 views

If $t \mapsto \lVert u(t) \rVert_{X}$ is measurable and square integrable then is $u \in L^2(0,T;X)$?

If $u:[0,T] \to X$ for a Banach space $X$ is such that $t \mapsto \lVert u(t) \rVert_{X}$ is measurable and $\int_0^T \lVert u(t) \rVert_X^2 $ is finite, does $u \in L^2(0,T;X)$? The usual definition ...