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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Prove $\mathcal{L}_p[0,1]$ is separable using Lusin and Stone-Weierstrass theorems.

1. Prove that the set of $p$-integrable functions on $[0,1]$ with the Lebesgue measure $\lambda$ is separable using Lusin's Theorem and the Stone Weierstrass Theorem. 2. Prove that ...
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35 views

What can we say about the inner product of two Cauchy sequences?

Let $(x_n)$, $(y_n)$ be two Cauchy sequences in an inner a real or complex product space $X$, and let the sequence $(\alpha_n)$ be given by $$ \alpha_n \colon= \ \langle x_n, y_n \rangle \ \ \ \mbox{ ...
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3 views

Operator norms and tensor norms on $M_n(A)$

If $A$ is a (unital) complex Banach algebra, then $M_n(A)$ can be equipped with the various operator norms (with respect to $p$-norms, say for $1<p<\infty$) and these are equivalent Banach ...
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39 views

Is the Inner Product a uniformly continuous function?

I know it's continuous but is it uniformly continuous?
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17 views

A question involving Frechet differentiability

Let $X, Y$ be real normed spaces and $U \subset X$ open subset. In "Nonlinear functional analysis and applications" edited by Louis B. Rall, we have the followint definition (page 115) A map $F : U ...
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33 views

Continuity of metric function

Let $X$ is a vector space and $d$ is a metric function on $X$ and $\|\cdot\|$ is a norm on $X$ and $\langle\cdot,\cdot\rangle$ is an inner product function on $X$ It is to easy to prove $\|\cdot\|$ ...
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1answer
23 views

Counter Example to Tietze Extension Property for Arbitrary Topological Space

Above is my question. My only issue is the final bit! For statements $1.$ and $2.$, the answer is true, since in both cases $Y$ is normal and we know that both metric and compact, Hausdorff spaces ...
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1answer
6 views

Reference request: Topological space of polygonal chains and its properties

I'm interested in approximations of $C^1([a,b])$-functions by polygonal chains: image File:NURBstatic.svg by User:WulfTheSaxon licensed under GNU Free Documentation License A polygonal chain can be ...
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1answer
6 views

Do the elements of a sequence converging to a point in the intrinsic core of a convex cone belong to the intrinsic core of the set eventually?

Let $X$ be a general Banach space and let $C\subset X$ be a convex cone. Consider a sequence $x_n$ in the affine hull of $C$ such that $x_n\to x$ for some $x\in icr(C)$, where $icr(C)$ denotes the ...
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1answer
28 views

Prove a subspace of a Banach space is closed

Let $X$ be a Banach space, $M$ and $N$ are two closed linear subspace of $X$. $N$ is finite dimensional.Prove $M+N$ is a closed subspace of $X$. It's trival to check that $M+N$ is a space. Then what ...
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18 views

Question about self adjointness

Let $A$ be a Hilbert space and $T:A\to A$ be linear and bounded. Is it in general true that $$T^*T \qquad \text{and}\qquad T+T^*$$ will both be self-adjoint? My intuition tells me the first one will ...
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16 views

Proof Check for Compactness of Integral Operator

Above is my question. I have completed the question, but I'm not 100% about my proof for the final part - it seems like I haven't done enough. I've shown that if $U$ and $V$ are compact, then so is ...
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1answer
20 views
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1answer
16 views

Isomorphism, Separable spaces

I am trying to show that: If there are sequences $(x_n)\subset X$ and $(y_n)\subset Y$ where $X$ and $Y$ are separable Banach spaces, such that $\overline{sp}\{x_n \mid n\in\mathbb{N}\}=X$ and ...
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31 views

Implications of Inner product vs Norm vs Metric Space

Is it true that: -an inner product satisfies the properties of a norm if and only if the norm satisfies the parallelogram equality -a norm can be induced by a metric if and only if the metric ...
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1answer
19 views

Prove that an infinite matrix defines a compact operator on $l^2$.

Let $(a(i))_{i=1}^\infty$ be an absolutely summable sequence, i.e., $\sum_{i=1}^\infty |a(i)|<\infty$, and consider the infinite matrix $$A=\begin{bmatrix} a(1)&a(2)&a(3)&\cdots\\ ...
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1answer
13 views

Analytic sets,, effros borel structure

Let SB denote the set of closed subspaces of $C(2^\mathbb{N})$ equipped with the Effros Borel structure, and $A\subset$ SB be analytic. I am reading a proof that says $A_\sim = \{Z\in $SB $ \mid $ ...
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1answer
25 views

Closed set in Baire space

I am reading a book on Banach spaces. It introduces the Baire space $\mathcal{N}=\mathbb{N}^\mathbb{N}$ as the product of infinitely many copies of $\mathbb{N}$ with the discrete topology. We have ...
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1answer
26 views

Is the relative interior of a subspace which is not closed empty?

In a general Banach space, the relative interior of a linear subspace which is not closed is empty, why ?
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1answer
40 views

A criterion for invertibility of a bounded linear map

Let $T$ be a bounded linear operator from the Hilbert space $H_1$ to a Hilbert space $H_2$. Suppose there exists $\delta>0$ such that $$ \langle T^{*}Tx,x\rangle\ge \delta \|x\|^2 $$ for all ...
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1answer
27 views

If $T^{*}$ is injective then $T$ is surjective?

If $T$ is a bounded linear map from the Hilbert space $H_1$ to the Hilbert space $H_2$, and $T^{*}$ is injective, then I know that $H_2$ is the closure of the range of $T$. But can I conclude that $T$ ...
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0answers
11 views

fourier transform of the indicator function [on hold]

Let $E$ be a discrete set in $\mathbb{R}$. If $E$ has the bounded density, then it is a tempered distribution. So I want to know if there is the exact formula of the fourier transform of the indicator ...
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1answer
15 views

Does every closed subspace of a dual space correspond to a closed subspace of its predual?

Suppose $X$ is a Banach space with dual space $X^*$. If $Y$ is a closed subspace of $X$, then $Y^\perp=\{x^*\in X^*: x^*(y)=0 \text{ for all } y\in Y\}$ is a closed subspace in $X^*$. I am wondering ...
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1answer
21 views

Characterization of invertibility of bounded linear operator between Hilbert spaces

Let $T$ be a bounded linear operator from the Hilbert space $H_1$ to a Hilbert space $H_2$. I've shown that the existence of a $\delta>0$ such that $$ \langle T^{*}Tx,x\rangle\ge \delta \|x\|^2 ...
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1answer
14 views

Injectivity of normal operators on a Hilbert space

Let $A$ be a bounded normal operator on a Hilbert space $H$. I know that $$ \ker A=(\text{ran} A^{*})^{\perp}. $$ What I've been unsuccesfully trying to prove is that $A$ is injective iff its range is ...
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1answer
30 views

Existence of double integral

the short time Fourier transform is obtained by the formula: $$Sf(u,\epsilon)=\int_\mathbb{R}f(t)g(t-u)e^{-i\epsilon t}dt$$ where $f,g \in L^2(\mathbb{R})$ are the signal and window respectively: ...
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24 views

Relationship between Cartan and Fréchet derivative

Let $f: X \rightarrow \mathbb{R}$ be smooth, then the Fréchet derivative is a map $Df: X \rightarrow L(X, \mathbb{R}).$ But if $f: M \rightarrow \mathbb{R}$ is smooth and $M$ a manifold, then the ...
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1answer
25 views

Proving subsets of $l^{\infty}$ are compact

Recently I started reading up on some set theory and metric spaces. I just read about compact subsets and I thought I understood it but in the exercises I'm having difficulty with the following ...
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71 views

Some questions about Krein-Rutman Theorem

I would like to figure out the Krein-Rutman Theorem. And I'm following the notes: ftp://ftp.ma.utexas.edu/pub/papers/llave/.grad/5999_chap1-1.pdf However, I got some questions. Defintion. Let X ...
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35 views

Can one define a functional on a Hilbert space based on its action on a Hilbert basis?

I know that the actions of a functional on a vector space can be uniquely described by the value the functional takes on each element of a (Hamel) basis. My question is, in a Hilbert space, would ...
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173 views

One of these two operators is not invertible

I have a Hilbert space $H$ and a bounded self-adjoint operator $T$ on it with $||T||=1$. I've been trying to show that at least one of $I+T, I-T$ are not invertible, but I haven't been able to make ...
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1answer
33 views

A characterization of Bessel sequences in a Hilbert space

I've shown that if for a sequence $\{f_{n}\}_{n=1}^{\infty}$ in a Hilbert space $H$ we have $$\sum_{n=1}^{\infty}|\langle f,f_n\rangle|^{2}< \infty$$ for all $f\in H$ (i.e., it is a Bessel ...
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2answers
34 views

Proving functional's continuity

Prove that functional $f:C[a,b]\to \mathbb{R},\ f(x)=\int_a^bx^2(t)dt$ is continuous. Any ideas on how to approach this problem?
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2answers
24 views

Proving mapping is contraction

Prove that mapping $B:C[0,\tau]\to C[0,\tau]$. $$(Bx)(t)=\left( \int_0^\tau \sin x(s)ds\right) t, \ t\in [0,\tau], \ \tau >0$$ is contraction mapping if $\tau^2<1$. I want to show that for all ...
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27 views

Existence of state on a C*-algebra satisfying $|\tau(ab)|=\|ab\|$ [on hold]

Let $a,b$ be elements of a unital C*-algebra $A$ with $0\leq a,b\leq 1$ (e.g., $a,b$ are projections). Is it the case there is a state $\tau$ on $A$ such that $|\tau(ab)|=\|ab\|$? If $ab$ is normal ...
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25 views

Wot topology on $B(H)$ is not metrizable

Let $H$ be a infinite dimensional Hilbert space and $B(H)$ be the space of bounded and linear operators on $H$. I know that weak operator topology (wot) and strong operator topology (sot) are ...
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5 views

Sufficient conditions on integration kernel for continuity of the integral operator

Suppose that we have a measure $d\mu(v)=e^{-|v|^2}dv$ on $\Bbb R^d$. We define a linear operator $$T[f](u)=\int_{\Bbb R^d} |u-v|^\beta d\mu(v).$$ I want to establish conditons on $\beta\in\Bbb R$ so ...
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1answer
34 views

Unitary G-module

I'm not sure if I understand this sentence correctly: "By a unitary G-module we will mean a Hilbert space W on which G acts by means of a strongly continuous unitary representation". $G$ is a ...
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1answer
18 views

Strong convergence of product of operators on a Banach space

If $\{T_n\},\{S_n\}$ are two sequences of bounded operators on a Banach space $X$, such that $\{T_n\}$ converges weakly to $T$, and $\{S_n\}$ converges strongly to $S$, does it follow that $T_nS_n\to ...
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1answer
18 views

Convergence in the weak operator topology implies uniform boundedness in the norm topology?

If $\{T_n\}$ is a sequence of bounded operators on the Banach space $X$ which converge in the weak operator topology, could someone help me see why it is uniformly bounded in the norm topology? I ...
2
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1answer
45 views

Show that this path is differentiable but not rectifiable

My path is defined as follows: $\gamma:[1,1]\rightarrow \mathbb R, \space \gamma(t):= \begin{cases} \ (0,0) & \text{if $t$=0} \\[2ex] t,t^2 \cos (\frac{\pi}{t^2}), & \text{if $t$ $\in$ ...
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1answer
41 views

What are the hypotheses in Levi's monotone convergence theorem?

Today I read monotone convergence theorem , dominated convergence theorem and fatou's lemma And I need some help We know the dominated convergence theorem in Measure theory In its proof we ...
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1answer
34 views

Convergence in Fréchet spaces and the topology

actually i have two questions concerning Fréchet spaces (i am not familiar with these spaces so i need some help). I have a vector space $V$ with a distance $$d(x,y) = \sum_{n\in \mathbb{N}} ...
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13 views

show there does not exist a best Approximation element in $E$ [on hold]

Let $c_{0}$ to be the space of sequence which converge to $0$,with $l^{\infty}$ norm,and $$ E=\bigg\{x=(x_{n})\in c_{0}\bigg| \sum_{n=1}^{\infty}\frac{x_{n}}{2^{n}}=0 \bigg\} $$ We know that $E$ is ...
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1answer
27 views

Is this sufficient for $f'' \in L^2$?

Let $f \in L^2(0,2\pi)$ be taken such that $f$ and $f'$ are absolutely continuous on $[0,2\pi]$ with $f(0) = f(2\pi)$ and $f'(0)= f'(2\pi).$ Is this sufficient to conclude from this that $f'' \in ...
2
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1answer
23 views

Normal Operators: Construction

Problem Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(T)\to\mathcal{H}:\quad N^*N=NN^*$$ Construct the operator: $$A:=N(1+N^*N)^{-1}\in\mathcal{B}(\mathcal{H})$$ ...
2
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0answers
31 views

A question regarding Parseval's identity.

In most books/websites, Proposition 2 (see below) is either stated for a Hilbert space or proved via Riesz-Fischer. Does the follow approach (which seems to work in an inner product space) fall down ...
2
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1answer
45 views

Identity of the Operator Norm applied to the differential $df$ for convex $U \subset \mathbb{R}^n$ and $f \in C^1(U, \mathbb{R}^k)$

In my Analysis II Script they often use 'special' norms such as the Operator norm to make proofs more 'elegant' or just shorter. There is also the following statement (without a proof) which I can't ...
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45 views

Showing that the sequence of functions is not Cauchy

I need to show that $ g_n(x)=x^{1/(2n-1)} $ is not a Cauchy sequence in $C[-1,1] $ w.r.t. supremum norm. I tried to find the maximum of the difference of $g_n$ and $g_m$ by just differentiating but ...
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1answer
49 views

A rather abstract strongly continuous semigroup

Define $X$ as the Hilbert space $L^{2}(0,\infty)$ and let the operators $T(t):X\to X$, $t\ge 0$ be defined by $(T(t)f)(\zeta):=f(t+\zeta)$ I want to show that $(T(t))_{t\ge 0}$ is a ...