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)

2
votes
1answer
67 views

Eigenvectors of a Self-adjoint Differential Operator Spans the Domain

Can someone, please, suggest a reference or what steps should I take to prove the following theorem: The set of eigenvectors of a self-adjoint differential operator, defined over a finite ...
1
vote
1answer
144 views

Counter example to an adaptation of the Riesz-Markov theorem.

Suppose that $(K,\tau)$ is a topological space and that $\phi$ is a positive linear functional on $C(K)$. Then is it true that there exists a unique Baire measure $\mu$ on $K$ such that $\phi(f) = ...
5
votes
0answers
161 views

How to decompose a representation into direct sum of cyclic representation?

Let $U$ be the bilateral shift operator on $l^2 (\mathbb Z)$, let $T=U+U^*$. How to calculate $\sigma(T)$? And how to show there is no cyclic vector for the action of $C^*(T,I)$. Further how to ...
1
vote
0answers
124 views

How can projection operators be limits of powers of unitary operators?

Consider a (fixed) unitary operator $U$ acting on the Hilbert space $\mathcal{H}$. Because the unit ball is compact in the weak topology, it is not hard to see that there exists a (smallest) compact ...
2
votes
1answer
123 views

Show for compact operator $K$, if $||Kf|| < ||f|| \forall f$, then $||K|| < 1$.

I wanted to check my reasoning on proving this statement, and see if anyone had suggestions for other proofs of this fact. Again, the statement is, if $K$ is a compact operator on a Hilbert space ...
1
vote
1answer
59 views

How to show $C(0,T;H)$ is Banach space

$H$ is a Hilbert space. How do I show that the Bochner space $C(0,T;H)$ of continuous $H$-valued functions is a Banach space with the following norm? $$\lVert u \rVert = \sup_{t \in [0,T]}\lVert u(t) ...
1
vote
0answers
47 views

All finite Baire measures are Closed-regular?

Given a finite Baire measure $\mu$ on a topological space $X$, is it true that $\mu$ is closed-regular? Where closed regular means that, $$\mu(A) = \sup\{\mu(K)| K \space \text{is a } Z\text{-set}, ...
5
votes
1answer
100 views

Are coordinate projections continuous?

Okay I have been working under the assumption that this is "obvious" for a while now, but it started to bug me and now I'm fumbling to prove it. Suppose $X$ is a normed linear space (possibly ...
0
votes
0answers
179 views

Can a function with finite discontinuities be nicely approximated by a continuous function?

1) Can every discrete function be approximated by some continuous function with regards to domain defined?? (By discrete function, I mean: there is countably infinite number in domain, and for each ...
1
vote
1answer
205 views

Continuous functional calculus

Let $\mathscr H$ be a Hilbert space, and $\mathscr B(\mathscr H)$ is a $C^*$-algebra, $T\in \mathscr B(\mathscr H)$ is a normal operator. Let $C^*(T)$ denote the $C^*$- subalgebra generated by $T$ ...
1
vote
0answers
61 views

Verify solution: Is this gradient, correct?

For a function $$f(X)=\operatorname{tr}(X^TAX)+\|\operatorname{diag}(X^TX)-\alpha I\|_2,$$ where all entries are real and $\alpha$ is a real scalar, while $A$ is a p.s.d matrix and $X$ is a real ...
0
votes
1answer
64 views

Proof of a special case of Banach's fixed point theorem

I have to prove the following special case of the theorem: Let $f : I \to I$ be Lipschitz continuous on the closed (not bounded) interval $I=[0,\infty)$ with Lipschitz constant $L \lt 1$. Then $f$ ...
0
votes
1answer
764 views

Find norm of the integral operator

Find norm of the following bounded linear operator $$Ax(t)=\int_0^1e^{-ts}x(s)ds$$ where $x\in C[0,1]$ and $t\in[0,1]$. Please help me.
3
votes
1answer
108 views

Existence of dominating measure for weak*-compact set of measures

Let $(\Omega,\mathcal F)$ be a measurable space and $\mathcal P$ a weak*-compact set of the set of all probability measures $\mathcal M_1(\Omega)$. Does there always exist a probability measure ...
0
votes
1answer
90 views

Linear operator

Is there a linear bounded (continuous) operator T from $c$ (convergent sequences with sup norm) ONTO $l^1$ (with its usual norm)? If it were so (which seems not), using the open mapping theorem we ...
1
vote
1answer
50 views

Is this function in the space $L^1$?

I have this function $$f(x)=\frac{1}{\vert x-y\vert^2(1+\vert x\vert^2)^s}$$ with $x\in\mathbb{R}^3$ and $y$ a fixed point. I have to study for which values of $s>0$ it belongs to ...
1
vote
0answers
123 views

From positive definite function to Følner sequence -— a question on amenability and nuclearity

We know that amenability of countable discrete group $\Gamma$ has many equivalent characterizations. In particular, there are two: a) there is a sequence of finitely supported positive definite ...
1
vote
1answer
58 views

Convergence in norm operator topology

I have to prove that a sequence $A(\varepsilon)$ of operators between Hilbert spaces $A(\varepsilon):H_1\to H_2$ converges, when $\varepsilon\to 0^+$, to an operator $B:H_1\to H_2$ in the uniform norm ...
3
votes
1answer
63 views

Block Matrices of Operators

I'm trying to prove the following: Consider the vector space of matrices of size $n\times n$ whose entries in $\mathcal B(H)$. Denote this vector space by $M_{n,n}(\mathcal{B(H)})$. We can define ...
2
votes
0answers
98 views

Counterexample for “the sum of closed operators is closable”

I'm looking for a counterexample in a Banach space. I've seen the counterexample at Sum of Closed Operators Closable?, but I don't understand why $A$ and $B$ are closed. Could someone expand on this ...
1
vote
0answers
45 views

What is the matrix norm in defining the generator of a continuous time Markov chain?

For a continuous time Markov chain with finite state space and Markov transition function $p(t)$, its generator $G$ can be defined entry-wise as $$ G_{i,j}:=\lim_{t\to 0^+} \frac{p_{i,j}(t) - ...
0
votes
0answers
92 views

All matrix/vector norms induce the same topology?

From Wikipedia all norms on $K^{m \times n}$ are equivalent; they induce the same topology on $K^{m \times n}$. This is true because the vector space $K^{m \times n}$ has the finite dimension $m ...
5
votes
1answer
94 views

Definite states on C*-algebras

A state $\omega$ on a unital $C^*$ algebra $A$ is called definite at $a\in A$ self-adjoint if $\omega(a^2)=\omega(a)^2$. I proved that if we have such a definite state at $a$, then for all $b\in A$ ...
1
vote
1answer
98 views

On the Spectral Theorem

Let $H$ be a Hilbert space, $T\in B(H)$ be normal and $E$ its spectral measure. a- Let $\delta >0$ , and let $M_{\delta}$ = $\left\{\lambda\in \sigma(T): |\lambda|\geq \delta\right\}$. ...
1
vote
1answer
433 views

Calculating the Norm of an operator in $L^2(0,1)$

If I have the following operator for $H=L^2(0,1)$: $$Tf(s)=\int_0^1 (5s^2t^2+2)(f(t))dt$$ and I wish to calculate $||T||$, how do I go about doing this: I know that in $L^2(0,1)$ we have that ...
0
votes
0answers
108 views

Showing an operator is self adjont

I am trying to show that the operator: $$Tf(s)=5s^2\int_0^1t^2f(t)dt+2\int_0^1f(t)dt$$ is self adjoint where $H=L(0,1)$ with real scalars and $t\in \mathcal{L}(H)$. So I can re-write this operator ...
5
votes
1answer
162 views

Continuity of Lebesgue integral with continuously-varying measures?

Consider the locally bounded mapping $m: X \times \mathcal{B}(X) \rightarrow [0,1]$, with $X \subseteq \mathbb{R}^n$ and $\mathcal{B}(X)$ denoting the Borel sets, such that for all $x \in X$, $\ ...
0
votes
1answer
103 views

Is it always true that the inner product is a map from a vector space to a scalar field?

In other words, is the inner product a bilinear functional? That is for $x,y \in V$ where $V$ is a vector space, is it always true that $\langle . \rangle: V\to \mathbb{F}$. Is it ever possible that ...
4
votes
1answer
186 views

States and positive elements in $C^*$-algebras

Let $A$ be a unital $C^*$-algebra and $w$ be a state (i.e a positive linear functional such that $\|w\|=w(1_A)=1$. I'm trying to prove the following:a) if $a$ is selfadjoint and $w(a^2)=w(a)^2$ then ...
4
votes
1answer
86 views

If $X$ has the discrete topology then the Stone-Čech compactification is zero dimensional.

If $X$ has the discrete topology then the Stone-Čech compactification is zero-dimensional. I would like to show the above to be true. If I can show that every closure (in $\beta X$) of an open set ...
0
votes
1answer
112 views

Bochner integral definition, I don't understand why this is true?

I have read something and I don'tunderstand why it is true!! Let $f \in L^2(0,t_0;X)$ where $X=L^2(\Omega)$ is Banach. By definition of the Bochner integral, there exists a seequince of measurable ...
7
votes
1answer
122 views

Is $\mathcal{C}([0,1])$ homeomorphic to a Hilbert space?

Let $\mathcal{C}([0,1])$ the Banach space of continuous functions from $[0,1]$ to $\mathbb{C}$. The norm on $\mathcal{C}([0,1])$ is $f \mapsto \| f\|_{\infty}= \sup_{x \in [0,1]} |f(x)|$. Is it ...
2
votes
1answer
78 views

A question concering nuclearity

B. Blackadar in his book Operator algebras: Theory of C${}^\ast$-Algebras and Von Neumann Algebras defines a C*-algebra $A$ to be nuclear if for every C*-algebra $B$ the algebraic tensor product ...
1
vote
2answers
83 views

Polynomials not dense in holder spaces

How to prove that the polynomials are not dense in Holder space with exponent, say, $\frac{1}{2}$?
2
votes
1answer
99 views

Is my proof correct? I want to show if $V \subset H$ is dense, then $L^2(0,T;V) \subset L^2(0,T;H)$ is dense too.

I want to show that if $V \subset H$ is a dense embedding then $L^2(0,T;V) \subset L^2(0,T;H)$ is dense too. Everything is a Hilbert space. Let $h \in L^2(0,T;H)$. Then $h(t) \in H$ for each $t$. By ...
3
votes
1answer
261 views

Sets $f_n\in A_f$ where $f_{n+1}=f_n \circ S \circ f^{\circ (-1)}_n$ and operator $\alpha(f_n)=f_{n+1}$

Let's start with a function on the Reals (in this case for $x=0$ is not defined): for example $f(x)=b/x$, $x \in \mathbb R$ I define: $$f_0:=f$$ $$f_{n+1}:=f_n \circ S \circ f^{\circ ...
1
vote
0answers
70 views

Is $C_0^\infty(0,T;H)$ dense in $L^2(0,T;H)$?

Let $H$ be a Hilbert space. Is the space $\mathcal{D}(0,T;H) = C_0^\infty(0,T;H)$ dense in $L^2(0,T;H)?$ I am aware that this space is dense in the space of differentiable Bochner functions but I ...
0
votes
2answers
113 views

Extending bilinear form from subspace to whole space

Let $X$ be a linear subspace of a Hilbert space $Y$. Let $a(\cdot,\cdot):X \times X \to \mathbb{R}$ be bilinear. Suppose I know what $a$ is on $X$. Is there some theorem or other that tells me that ...
2
votes
1answer
81 views

Doubt about Sobolev space norm

I consider the space $H^2(\mathbb{R}^3)$. I have a function and I have to verify that it belongs to this space. In the text I'm reading the author verifies that the function and its Laplacian are in ...
-1
votes
1answer
49 views

Norms and Finite Dimensions

In a finite dimensional spaces, is it true that if $\| \|_x > \| \|_y$ if $x > y$? So for instance is it true that $\| \|_4 < \| \|_6$ since $6 > 4$?
1
vote
0answers
95 views

Harnack's Inequality and (hypo)elliptic PDE

Background: I am aware of the Harnack's Inequality for linear elliptic equations. My questions are: (a) Is there a version of Harnack's Inequality for nonlinear elliptic equations, say, of the form ...
1
vote
1answer
37 views

On the weak closedness

I have some difficulties in this question. Let $X$ be a nonreflexive Banach space and $K\subset X$ be a nonempty, convex, bounded and closed in norm. We consider $K$ as a subset of $X^{**}$. I would ...
2
votes
0answers
45 views

Prove that the sequence is in $\ell^{2}$. [duplicate]

Let $(a_{n})$ be a sequence of complex numbers such that for every $(b_{n})\in \ell^{2}$the series $\sum_{1}^{\infty}a_{n}b_{n}$ converges. Prove that $(a_{n})\in \ell^{2}.$ What I've tried so far is ...
0
votes
1answer
56 views

Symmetric Operator with Different dot products

If I have a symmetric operator $A$ in a metric space $\mathscr{M}$. Then $\langle Au,v\rangle =\langle v,Au\rangle $ with the dot product defined in $\mathscr{M}$. My question is, if I keep the same ...
1
vote
0answers
49 views

Baire's theorem [duplicate]

I want to know interesting applications of Baire's Category Theorem. For example existence of no where differentiable function. Can any body tell me some similar applications?
1
vote
1answer
1k views

Inner product and norm of a function

I have recently started a undergrad. linear algebra course in which these definitions came up: Let $V$ be the vector space $C[a,b]$ of all continuous functions on $[a,b]$. Then the inner product and ...
6
votes
1answer
371 views

Maximal abelian subalgebra of Banach algebra is closed and contains the unit

I'm studying Murphy's book: C*-Algebras and Operator Theory, and got stuck on exercise 8 from chapter 1: "Show that if $B$ is a maximal abelian subalgebra of a unital Banach algebra $A$, then $B$ is ...
0
votes
2answers
214 views

Does the closed graph theorem presuppose that the domain is closed?

So this should be a very simple question. The Closed Graph Theorem as stated in Royden is Let $T\colon X \rightarrow Y$ be a linear operator between Banach spaces $X$ and $Y$. Then $T$ is continuous ...
2
votes
0answers
109 views

Which are nontrivial examples of analytical functions on Frechet spaces?

Let $X$ be a real linear topological space, which is a (separable) Frechet space, such that the topology on $X$ is generated by the countable family $\{p_n:n\in\omega\}$ of norms . A real-valued ...
2
votes
0answers
73 views

Tight Baire and Borel measures.

A Baire measure on a completely regular topological space $(X,\tau)$ is tight if, $$\mu(A) = \sup\{\mu(K)| K \space \text{is a compact} Z\text{-set}, K\subseteq A\}$$ A Borel measure on a completely ...