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
54 views

Do I have $\phi(a^*)=\overline{\phi(a)}$ for $\phi$ a State on a C*-Algebra?

Is the following correct or am I confused about something? $\phi(x^*)=\overline{\phi(x)}$ for $x\in A$ and $\phi\in S(A)$. Let $x\in A$, a C*-algebra. I can write $$x=a-b+ic-id,$$ where ...
2
votes
2answers
251 views

Applications of the Closed graph theorem

In functional analysis a famous theorem states that: if $X, Y$ are Banach spaces and $T: X \to Y$ is a linear operator, $T$ is continous if and only if the graph $\Gamma_T:={(x,Tx), x \in X}$ is ...
0
votes
2answers
52 views

Notion of convergence of equivalent norms is the same

I would like to make clear the proof for the following theorem which states that two norms over a vector space are equivalent iff their notion of convergence is the same. I have an hint for the proof ...
2
votes
2answers
612 views

Is the set of natural numbers with this metric complete?

Let $\mathbb{N}$, the set of all natural numbers, be given the metric $d$ defined as follows: $$ d(m,n) \colon= | m^{-1} - n^{-1} |$$ for all $m$, $m$ in $\mathbb{N}$. Then how to determine if ...
5
votes
1answer
109 views

Bounds on the line for entire functions of exponential type

Let $f$ be an entire function on the complex plane $\mathbb C$, assume that $$|f(z)|\le e^{|z|}.$$ Does the property $$|f(x)|\le e^{-|x|}, \qquad x\in\mathbb R,$$ imply $f\equiv 0$? More generally, ...
4
votes
1answer
913 views

Leray-Schauder fixed point theorem

I know the proof of the Schauder fixed point theorem which states Schauder fixed point theorem : If $D$ is a non-empty , convex and compact subset of Banach space $B$ and $T:D \to D$ a ...
1
vote
2answers
348 views

Is this metric space complete?

Let $a$, $b$ be two real numbers such that $a<b$, and let $X$ be the set of all (real or complex-valued) functions defined and continuous on $[a,b]$ with the metric $d$ defined as follows: $$ ...
0
votes
2answers
67 views

Erwine Kryszeg's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS, Section 1.5-8

In Section 1.5-8, in his book, INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS, Kryszeg tries to show that the set $X$ of all polynomials defined on a given closed interval $[a,b]$ on the real ...
0
votes
1answer
56 views

Show that the set of uniformly Lipschitz functions vanishing at $0$ is compact in $C[0,1]$

The question is: For $K$ and $\alpha$ fixed, show that $\{f\in \operatorname{Lip}_k \alpha : f(0) = 0\}$ is a compact subset of $C[0,1]$. I was going to attempt this by using by Arzela-Ascoli theorem ...
2
votes
1answer
54 views

continuous functional calculus; spectrum of an self adjoint element in a c*algebra

Let A be a C$^*$-Algebra, $a\in A$ selfadjoint and $\|a^2-a\|<\frac{1}{4}$. The claim is: $\sigma(a)\subseteq (-\frac{1}{2},\frac{1}{2}) \cup (\frac{1}{2},\frac{3}{2})$ and there is a projection ...
3
votes
1answer
40 views

Is it true that $\|A+PBP\|\le\|A+B\|$ for every projection $P$ and positive operators $A,B$?

Let A and B be positive operators on and let P be a projection. Is the inequality $$\|A+PBP\|\le\|A+B\|$$ true? Here $\|.\|$ stands for the operator norm.
1
vote
2answers
32 views

Show that the function 1/t is not in L2 (0,1]

Need some help getting started with this problem: $$f(t) = \frac{1}{t}$$ Show that $f(t)$ is not in $L_2(0,1]$, but that it is in the Hilbert space $L_{2}w(0,1)$ where the inner product is given by ...
1
vote
1answer
71 views

Showing space of functions with lipschitz norm is complete

I have a Banach space, $X$, given by all the complex valued functions $x: [-1,1] \to \mathbb{C}$ where $x(0) = 0$. And I've shown that the following defines a norm on $X$: $$\|x\| = inf \{ \beta : ...
2
votes
0answers
28 views

Finding a norm on $ \mathbb{R}^X $ such that the “natural” embedding of a metric space $ X $ in $ \mathbb{R}^X $ becomes an isometry

This question is related to this one: Every metric space can be isometrically embedded in a Banach space, so that it's a linearly independent set Kuratowski's embedding indeed seems to be the ...
1
vote
0answers
60 views

How to prove the following isomorphism?

Let $A, B$ be two C*-algebras, $\pi:B\rightarrow A$ and $\sigma: A\rightarrow B$ be *-homomorphisms such that $\sigma\circ\pi$ is homotopic to $1_{B}$. Define a *-homomorphism $\delta: B\rightarrow ...
0
votes
1answer
52 views

Sequence of unitary l.i. vectors such tha the sequence converges weakly to a non-zero vector, but not strongly

Let $\mathcal H$ be an infinite dimensional Hilbert space and let $\{x_{n}\}_{n=1}^{\infty}$ be a sequence of unitary linearly independent vectors. I know, using Bessel's inequality, that if the ...
0
votes
1answer
31 views

A question about finite-rank projection

Let $B, C$ be two C*-algebras and $\sigma_{0}: B\rightarrow C$ be *-homomorphism such that $\sigma_{0}$ is injective. Then, for a finite set $F\subset B$ of the unit ball and $\varepsilon>0$, Can ...
1
vote
1answer
67 views

p is a projection iff p is normal the spectrum of p is contained in {0,1}

I want to know why the following claim is true: Let A be a C$^*$-Algebra. $p\in A$ is a projection (that means $p^2=p^*=p$) iff p is normal and $\sigma (p)\subseteq \{0,1\}$. "=>" why p normal, it is ...
5
votes
1answer
99 views

Existence of a mapping in a nonseparable Banach space that moves all nearby points to far-away points

Does there exist a nonseparable Banach space $X$, a mapping $F: X\to X$, and an open nonempty subset $D\subset X$ such that $$ \forall\,E>0 \quad \exists\,\delta>0: \quad \forall\,x,y\in D \quad ...
0
votes
1answer
71 views

A question about finite-rank projection on Hilbert space

Let $H$ be a Hilbert space and $P_{n}\in B(H)$ be an increasing net of finite-rank projection which converge to the identity in the strong operator topology. Then, Can we verify that ...
1
vote
0answers
38 views

Equivalence discrete H^2 Sobolev norms

My aim is showing the equivalence of two discrete Sobolev norms. On $\mathbb{Z}^d$, $d\ge 2$, one defines the discrete derivative in the direction of the coordinate vector $\vec e_j$ as $$ ...
1
vote
2answers
285 views

Bounded measurable functions

Suppose $X$ is a compact space and $B(X)$ denotes the bounded Borel measurable function space. Let $f\in B(X)$. There is a sequence of step functions $\{\phi_n\}$ such that $\phi_n\to f$ (point wise). ...
0
votes
1answer
133 views

Arzela-Ascoli equivalent theorems

The following theorems are equivalent? Is the Theorem 2 false? Theorem 1 (Arzela-Ascoli): Let $X$ be a compact metric space and let $C(X)$ denote the space of continuous functions $f: X \to \mathbb ...
0
votes
1answer
38 views

Open Mappinig Theorem doubts on Brezis's Book

In famous book of Brezi's book of Functional Analysis that can be found here. I have some doubts on the also famous open mapping theorem in his first step on page 36, where he wants to prove that if ...
0
votes
1answer
64 views

Proof about distances in banach spaces

Let $(X,\|\circ\|)$ be a banach space with $$\forall x,y\in X, x\neq y, \|x\|=\|y\|=1 \Rightarrow \|\frac{x+y}{2}\|<1 $$ If M is convex and $z \in X$, then $\|x-z\|=dist(z,M)$ for at most one $x ...
1
vote
1answer
63 views

A lemma about quasicentral-approximate-unit

Here is a lemma about quasicentral-approximate-unit: Lemma 7.3.1Let $J\triangleleft A$ be a separable ideal. Then there exists a quasi-central approximate unit $\{e_{j}\}\subset J$ such that ...
2
votes
1answer
139 views

Resolvent also self-adjoint operator

If I have a self-adjoint operator $U : \operatorname{dom}(U) \subset H \rightarrow H$ and $\lambda \in \rho(U)$, then I assume assume that it is correct that the operator $(U - \lambda I)^{-1} \in ...
2
votes
1answer
51 views

QD C*-algebra's representation theorem

Here is a question from the proof of the "QD C*-algebra's representation theorem" in P245 of book "C*-algebras and Finite-Dimensional Approximations" by Nate and Taka. For a separable unital ...
1
vote
1answer
62 views

A question about spectral theorem

The following is a discussion about spectral theorem of Folland's Harmonic analysis page 18. Suppose $A$ is a unital commutative C*- subalgebra of $B(H)$ and $u,v\in H$. Put $\Sigma = \sigma(A)$ . ...
6
votes
2answers
949 views

Prove that this integral operator is compact

Let $X,Y=L^2(0,1)$, $k\in C^0([0,1]^2)$. Define $$ K:X\to Y,\,\,\,\,\,Kf(x):=\int_0^1k(x,y)f(y)dy\,\,\,\,\forall\, f\in L^2(0,1). $$ I have to show that $K$ is compact. My idea is to prove that $K$ ...
2
votes
1answer
64 views

Hypothesis of Riesz's representation theorem

The Riesz representation states that Let $(H, \langle \cdot, \cdot \rangle)$ be a Hilbert space. Then for any linear and continuous functional $x^{\ast}:H \to \mathbb{K}$ there is a unique element ...
0
votes
1answer
35 views

Computing adjoint of a linear operator

I would like to know how to find an adjoint of an operator $T$ on a Hilbert space. I tried to find out on my own but it's not solid. Here is what I did: I picked a concrete example. Let $H=\ell^2$ ...
2
votes
1answer
72 views

Operator classification

Imagine that we have a second-order Sturm-Liouville problem on an interval $(a,b)$. What is the relationship between the structure of solutions and the dimension of $\ker(T^* \pm i)$, does anybody ...
1
vote
0answers
16 views

Nonlinear continuous and unbounded operator [duplicate]

Let $X$ be an infinite-dimensional Banach space, and let $B=\{x\in X: \|x\|\leq 1\}$ be its closed unit ball. Does there exist a continuous mapping $F: X\to X$ such that the set $F(B)=\{F(x): x\in ...
2
votes
1answer
97 views

Weak convergence in Banach spaces

In Rudin's book <> Page 66, it says "If $X$ is a infinite dimensional topology vector space, then $X$ under the weak topology is not locally bounded" . Hence I think the topology of any ...
3
votes
1answer
152 views

Generalised derivative and derivative of functions of bounded variation

Let $f:\mathbb{R}\to\mathbb{C}$ be a function Lebesgue-integrable on any finite interval and let $K$ be the space of infinitely differentiable equal to 0 outside a given finite interval. Be the ...
2
votes
2answers
50 views

Finding simplest function to distinguish 2 sets

I wish to find a function that distinguishes $2$ sets. I have m data values in form of n-tuples out of which k are supposed to be mapped to a value less than $0$ and other m-k are supposed to be ...
0
votes
1answer
45 views

Is the monotone convergence theorem bidirectional?

Say I have $(f_n)$ with $f_1 \le f_2 \le ...$ and I know that $\lim_n\int f_n<\infty$ exists, does that imply $f_n$ converges a.e.? Most formulations I have seen of the monotone convergence ...
2
votes
0answers
70 views

James $\ell_1$-theorem: problem in the proof

I am struggling with the very last estimate in the proof of James' $\ell_1$-theorem. (Please see below an excerpt from Albiac and Kalton's fantastic book Topics in Banach space theory.) I don't ...
1
vote
1answer
54 views

Power series for functionals and notation for functionals

I am trying to learn some functional analysis/calculus of variations, mainly for being able to perform functional derivatives on simple functionals found in physics (therefore I will not be too ...
1
vote
0answers
60 views

Alternative proof of "Every linear mapping on a finite dimensional space is continuous”

Here is my question: Suppose that $T:X\to Y$ is linear, where $X$ and $Y$ are normed linear spaces, and $X$ is finite dimensional. Define $\|\cdot\|_\beta$ on $X$ by ...
8
votes
1answer
200 views

Use $C^\infty$ function to approximate $W^{1,\infty}$ function in finite domain

This is exercise 10.21 from Leoni's book. The exercise asks me to prove that for any $u\in W^{1,\infty}(\Omega)$ where $\Omega$ is open FINITE, there exists a sequence $(u_n)\subset C^\infty(\Omega)$ ...
1
vote
2answers
56 views

$X$ is the vector space $C[0,1]$ with the norm $\|f\|_1=\int_0^1|f(t)|dt$, and $M=\{f\in X:f(0)=0\}$, show that $M$ is not closed.

Here is my question: Let $X$ be the vector space $C[0,1]$ with the norm $\|f\|_1=\int_0^1|f(t)|dt$. Let $$M=\{f\in X:f(0)=0\}$$ Show that $M$ is not closed. Show that the “quotient norm" ...
2
votes
1answer
84 views

$X=C[0,1]$ is a Banach space, $M=\{f\in X: f(0)=0\}$, prove $M$ is closed, find explicit formula for the quotient norm, and find an isomorphism.

Here is my question: Let $X$ be a Banach space $C[0,1]$ with the supremum norm. Let $M=\{f\in X: f(0)=0\}$. Show that $M$ is closed. Find an explicit formula for the quotient norm $\|[f]\|$ for ...
1
vote
1answer
43 views

Show that $\lbrace S_n x \rbrace$ converges for a particular recursively-defined sequence of operators $S_n$

$H$ is a Hilbert space, $M$ is a self-adjoint bounded linear operator on $H$ with $M \leq I$, and $S_0 = 0$; $S_{n+1} = (1/2)(M + S^2_n)$ for $n = 0, 1, 2, ...$. For all $n$, both $S_n$ and $S_n - ...
0
votes
2answers
49 views

If $\sum_{n=1}^\infty \|x_n\|\lt\infty$, , then $\lim_{k\to\infty}\sum_{n=1}^k x_n$ exists

Here is my question: Let $X$ be a Banach space with norm $\|·\|$. Prove that, for any sequence $\{x_n\}$ in $X$, if $\sum_{n=1}^\infty \|x_n\|\lt\infty$, then $\lim_{k\to\infty}\sum_{n=1}^k x_n$ ...
1
vote
1answer
101 views

Generalised derivative of Cantor staircase

If we consider the Cantor staircase function, let us say $f:[0,1]\to\mathbb{R}$, as a distribution, I was wondering whether there is an explicit way to express its generalised derivative as a ...
1
vote
1answer
29 views

Verifying a bound on the norm of an operator in $l_2$.

The problem: Define $L: l_2 \rightarrow l_2$ by $L(x_1, x_2, ...) = (y_1, y_2, ...)$, where $y_n = (x_1 + x_2 + ... + x_n)/n^2$. Show that $||L|| \leq (\sum_{n=1}^\infty 1/n^2)^{1/2}$. My proof: ...
1
vote
2answers
37 views

Equivalance of norms

Let $X$ be the vector space of all real valued functions defined on $[0,1]$ having continuous first-order derivatives. How to show that the following norms are equivalent: $\|f\|_1 = |f(0)| + ...
-1
votes
1answer
43 views

How can I complete my proof: Sobolev space W^(1,p) is complete? Using Convergence theorem

I'm trying to prove that W^(1,k) (R) is complete. The steps i Had so far: let {fn} be a cauchy sequence in W^(1,k). therefore {fn} and {dfn} are cauchy sequences in L^p(R), and therefore converge ...