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, ...

learn more… | top users | synonyms (1)

1
vote
1answer
38 views

If X be a Hilbert space, How can write above condition?

For points $p,q,r,s\in X$, $$(d(r,p)‎\leq‎ d(p,s)\ \&\ d(r,q) \leq‎ d(q,s))\Rightarrow‎ d(r,m)\leq‎ d(m,s)$$ for any point $m$ in the segment $[x,y]$. \How is this condition for Hilbert spaces?
1
vote
1answer
35 views

what does the mapping in this exercise for uniform convergence mean

Let $X,Y$ be normed vector spaces. Show: If $A_n(t)\in L(X,Y)$ converges strongly and uniformly for $t\in [0,1]$ and if $x\in C^1([0,1],X)$, then $A_n(t)x(t)$ converges uniformly for $t\in [0,1]$. I ...
1
vote
1answer
72 views

If $\|T\| < 1$, then $I-T$ is invertible and $\|(I-T)^{-1}\| \leq (1-\|T\|)^{-1}$

This is a hint in my functional analysis book, and I can`t uncipher it. They give as additional information that $T \in B(X)$ where $X$ is a normed linear space. I think $X$ should be a Banach Space, ...
13
votes
2answers
225 views

Is there a concept of a “free Hilbert space on a set”?

I am looking for a "good" definition of a Hilbert space with a distinct orthonormal basis (in the Hilbert space sense) such that each basis element corresponds to an element of a given set $X$. Before ...
2
votes
0answers
34 views

The adjoint of unbounded operators as a function.

Let $H_1$, $H_2$ be two possibly distinct real or complex Hilbert spaces, with linearity in the first coordinate of the inner product for concreteness. Let's think of passage to the adjoint as a map ...
1
vote
0answers
63 views

Extension of a Pseudodifferential Operator

Let $M$ be a smooth manifold with countable atlas, and define the distributions $\mathscr{D}'(M)$ as the dual space to the smooth densities with compact support, and $\mathcal{E}'(M)$ as the dual ...
1
vote
2answers
32 views

Boundedness in $(M_{\phi}f)(x)=\phi(x)f(x)$ and $(Kf)(x)=\int_a^bk(x,t)f(t) dt$

I'm currently practicing (for a comprehensive exit exam) how to prove something is bounded. Here are 2 questions I'm concerned about: Let $\phi$ be a continuous function on the interval $[a,b]$. ...
0
votes
1answer
97 views

Is operator $T_n$ a compact operator?

Is operator $T_n$ a compact operator? $$T_n:l_2\rightarrow l_2$$ $$T_nx=(\underbrace{0,0,\ldots,0,}_{n\text{ zeros}}, x_1,x_2,x_3,\ldots)\text{ where }x=(x_1,x_2,x_3,\ldots)\in l_2,\ \sum^\infty_{k=...
1
vote
1answer
127 views

Transpose of a differential operator

Let $H$ be a diagonalizable matrix (not necessarily Hermitian). Then, it induces a biorthogonal left and right vectors, such that $$ H\left|\lambda\right\rangle=\lambda\left|\lambda\right\rangle,\quad\...
3
votes
1answer
95 views

Non-existence Tracial states

We know that every non-zero finite dimensional C*-algebra has a tracial state. I am searching for an example of a simple C* algebra without tracial state with explaination. I think you have to look to ...
2
votes
1answer
30 views

Estimation with an orthonormalbasis in some finite dimensional subspace of $L_2(\Omega)$

I'm currently trying to understand a step of a proof of the following estimation: $\displaystyle \sum_{j=2}^{N} \left(\left< \varphi_1-R_N\varphi_1, \varphi_{j,N} \right>_{L_2}\right)^2 \ \...
1
vote
1answer
148 views

Decay of a Convolution

Let $f, g \in L^1\cap L^\infty(\mathbb{R}^d)$ be probability distributions on $\mathbb{R}^d$, and suppose at large $|x|$, $f$ decays like $|x|^{-\alpha}$ while $g$ decays like $|x|^{-\beta}$, with $\...
4
votes
2answers
80 views

Operators $A$ such that $e^A$ is norm preserving

Let $X$ be a Banach space. $A$ a bounded operator. We can define the exponential of $A$ by $$e^{A}=\sum_{n=0}^{+\infty}\frac{A^n}{n!},$$ which is also a bounded operator. Is there any sufficient ...
2
votes
1answer
54 views

Dense subspace of linear space and functional equal to $0$

Let $X$ be a normed space over field $\mathbb{K}$ and let $X_0$ be a linear subspace of $X$. I have to prove that: $X=\overline X_0 \iff$ For every linear continuous functional $\phi : X\rightarrow \...
2
votes
1answer
66 views

Two questions about a proof of the compactness of an operator

There are a few things that I don't understand about a proof and I'd appreciate any help. The theorem and its proof are the following: (1) Is the equality $$ \|v(\tau) -v(\tau_j)\| = \max_{1 \le ...
1
vote
0answers
56 views

Let $T$ be a bounded operator such that $<Tf,f>=0$ then $T=0?$

Let $H$ be a hilbert space. Let $T:H\to H$ be a linear bounded operator such that $<Tf,f>=0$ for all $f\in H$. It is necesarily true that $Tf=0 ?$ When I mean Hilbert space over a field $\...
0
votes
1answer
32 views

Set of polynomials

I want clarification on the following question: Let $\{c_0,c_1,c_2,\dots,c_n\}$ denote a set of $n+1$ distinct elements in $\mathbb{R}$. Define the set of $n+1$ polynomials. $$f_j(x)=\prod_{k=0,k\ne ...
1
vote
2answers
132 views

Fourier series with respect to orthonormal sequence

Let $H$ be the space of piecewise continuous $2 \pi$-periodic functions on the real line. For $f$ and $g$ in $H$, consider the inner product $<f,g>=\frac{1}{2\pi}\int_{- \pi}^{\pi}f(x)\overline {...
1
vote
3answers
66 views

How to find the spectrum?

Let the Hilbert space $H=l_2$ over the complex field. How to find the point spectrum $\sigma_p(A)$ of $A$ $Ax=(x_1,ix_2,-x_3,-ix_4,x_5,....)$ Any help is very appreciated. thanks :)
0
votes
3answers
190 views

Does $L^2$ strong convergence and bounded $L^\infty$ imply convergence in $L^\infty$ along subsequences?

Take a sequence of functions $f_n \in L^2([0,1])$ such that $f_n \rightarrow f$ in $L^2([0,1])$ (convergence in $L^2$ norm) and $|f_n|_{L^\infty},|f|_{L^\infty} <R$ (uniformly bounded in $L^\...
2
votes
0answers
47 views

$L^{\infty}(\mu)$ is $C(K)$ for some compact Hausdorff space $K$ when $\mu$ is sigma-finite

I am looking for a proof (or at least an article or book in which it is stated) that for measure space $(X, A, \mu)$, where $\mu$ is sigma-finite, there exists compact Hausdorff space K such that $L^{\...
1
vote
1answer
1k views

Counterexamples for L1-convergence not implying L2-convergence and vice versa

I am trying to find counterexamples for the following statements. Let $\{f_n\}$ be a sequence in $L^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d)$, and let $f$ also be in $L^1(\mathbb{R}^d) \cap L^2(\...
0
votes
1answer
113 views

Prove that the shift operator $(Ax)(t)=x(t−a)$ not compact.

Prove that the shift operator $(Ax)(t)=x(t−a)$ not compact in the space of bounded continuous functions on R.
0
votes
1answer
24 views

Is there any linear isomoprhism between $X'$ and $Y'$?

Let $X$ and $Y$ be Banach spaces, and $T: X\rightarrow Y$ be an linear isomorphism. Is there any linear isomoprhism between $X'$ and $Y'$? Any help?
1
vote
3answers
109 views

Finding spectrum of the operator A

$A:\mathcal{l}_2\rightarrow \mathcal{l}_2:(x_n)_{n=1}^\infty \rightarrow (x_{n+1})_{n=1}^\infty$ (left shift) Find the spectrum and all its parts for the operator A. What should I do?
1
vote
1answer
57 views

How to show whether this operator is normal? self-adjoint? unitary?

Consider the Hilbert space $H=l_2$ over the complex field and $A:H\rightarrow H.$ How to show whether this operator is normal? self-adjoint? unitary? $A(x)=(x_1,0,0,\frac{1}{2}x_4,0,0,0,0,\frac{1}{3}...
1
vote
1answer
40 views

Sturm Liouville problem with additional term.

Imagine you want to solve an ODE on $[a,b] \subset \mathbb{R}$ $f''(x) + (A(x) + B(x))f(x) = \lambda_n f(x)$, where $A,B$ are some smooth functions and $\lambda_n$ the n-th eigenvalue. Furthermore, ...
1
vote
1answer
33 views

Is strong convergence?

Let $F_n(f) = f(\frac{1}{n}) - f(-\frac{1}{n}) \in \big ( C^{(1)} [-1, 1] \big ) ^*$. For every $f \in C^{(1)} [-1, 1]$ we have pointwise limit: $$ \lim_{n \to \infty} F_n (f) = \lim_{n \to \infty} \...
1
vote
1answer
29 views

How to show that the spectrum is equal to the range of $y$

How to show that the spectrum of $T_y$ is equal to the range of $y$ Given $y\in C[0,1]$ and $T_y: C[0,1] \rightarrow C[0,1]: x\mapsto x\cdot y$ Any help is appreciated, thanks.
2
votes
1answer
43 views

Duals of embeddings in the space of distributions

If $ \Lambda \colon X \hookrightarrow \mathcal{D}'$ is a continuous embedding of a normed vector space $X$ into the space of distributions (for example $X=L^p$), is it true that the dual of $\Lambda(X)...
2
votes
2answers
78 views

About a spectrum of a C*-algebra

Let $A$ be an unital commutative C*-algebra. Show that the spectrum of $A$ is disconnected iff there is a projection $p \in A$ not trivial.
2
votes
1answer
216 views

Projection onto finite-dimensional subspace of $L^p$

Let $a_i$ be a basis of $L^p(\Omega)$ and consider $A_n = \text{span}\{a_1, ..., a_n\}$. Take an element $f \in L^p$. We want to define a projection onto the finite-dimensional subspace $A_n$. How do ...
0
votes
1answer
51 views

derivate of indicator function [on hold]

What is the derivative of the indicator function: \begin{equation} f(x)=\begin{cases} 1 & x^{\min} x\leq x^{\max}\\ -\infty &\mbox{otherwise}? \end{cases} \end{equation} thank you
0
votes
1answer
82 views

quotient map surjective and linear

Let $Q:X\to Y$ be a quotient map, i.e. a map between normed spaces such that $B_Y(0,1)=Q(B_X(0,1))$ ($Q$ maps the unit balls onto each other). Then $Q$ is surjective and $Q\in L(X,Y)$ with $\|Q\|=1$. ...
0
votes
0answers
34 views

Hausdorff topologies on a finite dimensional vector spaces

Let $X$ be a finite dimensional vector space (over complex numbers), $\mathcal{P}_1$,$\mathcal{P}_2$ - two families of seminorms. There is the following statement: these families are equivalent if the ...
1
vote
1answer
80 views

Problems proving that a compact operator is completely continuous [duplicate]

I would like to prove that if $T:X\rightarrow Y$ is a compact operator, then for every weak convergent sequence $(x_n)_{n\in\mathbb N}$ with $x_n\rightharpoonup x$ for some $x\in X$ it follows that $...
0
votes
1answer
40 views

Point-wise convergence cannot be normed

Let $X$ be an arbitrary set. Consider the space $\mathbb{C}^X$ of all functions $X\to \mathbb{C}$. For each $x\in X$ we build a seminorm $||\cdot||_x$ such that $||f||_x=|f(x)|$. I would like to prove ...
0
votes
1answer
43 views

The hereditary subalgebra

If $B$ is a C*-algebra and $A\subset B$ is a hereditary subalgebra, then , taking $\{e_{n}\}$ be the approximate unit of $A$, can we verify $e_{n}be_{n} \in A$ for every $b\in B$?
0
votes
1answer
83 views

The point-ultraweak convergence of contractive completely positive map

Let $A$ and $C$ be C*-algebras. If $\phi_{n}: A \rightarrow C$ is a c.c.p (contractive completely positive) map, then the point-ultraweak cluster point of the map $\phi_{n}$ is still a c.c.p. map? Why?...
1
vote
1answer
56 views

Completely bounded map and minimal tensor products

Theorem 3.5.2. Let $\phi: A\rightarrow C$ and $\psi: B\rightarrow D$ ($A, B, C, D$ are C*-algebras) be c.p.(completely positive) maps. Then the algebraic tensor product map $$\phi\odot\psi: A\odot ...
3
votes
1answer
130 views

Compact operator as a limit of finite ranked operators

So here is my question, I had to show that the following operator is compact, $$T:C[0,1]\rightarrow C[0,1]$$ $$f\mapsto\int_0^tf(s)ds$$ with $||f||=\mathrm{sup}_{x\in[0,1]}|f(x)|$ I think I ...
0
votes
1answer
56 views

Exercise on isometry

Let $X$ be a Banach space and $T$ a linear bounded operator defined on $L(X,Y)$ with $Y$ a normed space. If $T$ is an isometry then $TX$ is a closed subspace of $Y$. I considered a sequence $y_n$ ...
2
votes
1answer
108 views

Diagonal convergence

Let $(x_r)$ be a bounded sequence of points in $X$ and $(f_n)$ be a sequence of functions on $C(X)$ convergent pointwise to $f \in C(X)$. And so for each point $x_r$, the sequence $(f_n(x_r))$ ...
0
votes
0answers
29 views

When can we get discrete spectrum?

Suppose that $T$ is a densely defined closed operator on a separable Hilbert space $H$. Form $N = T^*T$. Assume further that $T$ has a finite dimensional kernel and satisfies the commutation relation $...
0
votes
1answer
76 views

If the composition of 2 maps is open and one of 2 maps is open, then so is the other map.

From "Functional analysis" by Walter Rudin, page 39, exercise 9 (and replace "TVS" with "normed vector space"). Suppose (a) $X$ and $Y$ are normed vector spaces, (b) $\Lambda \colon X \to Y$ is ...
1
vote
1answer
54 views

questions about $L^p$ space with $0<p\leq 1$ parallel to the case $1<p$

Question (1). Riesz-Fischer Theorem: For $1\leq p\leq \infty$, $L^p(\mu)$ is complete. Corollary of proof: Let $1\leq p\leq \infty$. If $(f_n)_{n=1}^\infty$ is a sequence coverging to $f$ with ...
0
votes
1answer
65 views

Bounded Linear Transformation proof

One paragraph in my text is to prove that $\|T\|=\sup\{|\langle Tf, g\rangle|:\|f\|<1, \|g\|<1\}$, where we have a bounded linear operator between two Hilbert spaces $T:\mathcal H_1\rightarrow \...
1
vote
1answer
37 views

Unbounded spectrum can be empty

Can someone please provide an example of a Hilbert space $H$ together with a (prefer densely defined if possible) unbounded operator $T$ such that $\sigma(T)$ is the empty set? I have tried to use ...
1
vote
1answer
37 views

When $A_y$ is invertible?

Given $y\in C[0,1]$ Let $A_y:C[0,1]\rightarrow C[0,1]: x\mapsto xy$ When $A_y$ is invertible? Could you please help.
0
votes
1answer
54 views

Inner product on Hilbert Spaces

It's an open question. How could you define an inner product for a product of noncontable Hilbert spaces?