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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4answers
66 views

Solving a functional equation ( $ f(x-y) = f(x)/f(y)$ )

Consider the functional equation $$f(x-y)=f(x)/f(y)$$ If $f'(0)= p$ and $f'(5)=q$, then what is the value of $f'(-5)$ ? My attempt. Using the equation written above I was able to determine the ...
2
votes
0answers
10 views

Universal $C^*$-algebra of countable family of self-adjoint operators have boundedly complete standard Schauder basis

Let $A = C^*(T_1,T_2,T_3,... | T_i=T_i^*, ||T_i|| \leqslant 1)$ - universal enveloped $C^*$-algebra of countable family of self-adjoint operators. $A$ have standard Schauder basis, which contains all ...
0
votes
1answer
787 views

Show that for any partition $P$ of $[a,b]$, $U(f,P) - L(f,P) \leq C(b-a)mesh(P)$.

Suppose $f:[a,b]\to \mathbb{R}$ is Lipschitz, i.e $|f(x)-f(y)| \leq C|x-y|$ for all $x,y$ in $[a,b]$ and thus $f$ is continuous. Show that for any partition $P$ of $[a,b]$, $U(f,P) - L(f,P) \leq ...
5
votes
1answer
97 views

Connection of Fourier's work with Fredholm's

Im trying to formulate for myself in what sense Fredholms work on the Dirichlet problem is connected to Fouriers work on the heat equation. Fourier idea seems to have fundamental problems with ...
9
votes
1answer
2k views

Proof of separability of $L^p$ spaces

The following is a proof in Brezis book. It shows the separability of $L^{p}$ spaces: I have a few questions regarding the proof: It says 'it is easy to construct a function $f_{2} \in ...
0
votes
0answers
25 views

Is there a way to derive/prove that $ \langle Ux, t \rangle = \langle x, U^{-1}t \rangle$ is true for unitary transformations?

Why is it that $ \langle Ux, t \rangle = \langle x, U^{-1}t \rangle$ is true for unitary transformations? I have a argument (not quite a proof I guess) of why that is true for finite vectors but I ...
-1
votes
0answers
12 views

How do I find not empty level sets of function?

Let $f$ be defined as follows: $$f:\mathbb{R}^{2}\to\mathbb{R}:(x,y)\mapsto\begin{cases}\frac{xy^{2}}{x^{2}+y^{4}}&\text{if } (x,y)\neq (0,0)\\ 0&\text{if } (x,y)=(0,0)\end{cases}$$
0
votes
0answers
12 views

If $Q$ is a trace class operator on $U$, then each bounded, linear operator from $U$ to $H$ is a Hilbert-Schmidt operator from $Q^{1/2}U$ to $H$

Let $U$ and $H$ be Hilbert spaces $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ with finite trace $U_0:=Q^{1/2}U$ $L$ be a bounded, linear operator from $U_0$ to $H$ I ...
3
votes
1answer
46 views

Is the derivative of differentiable function $f:\mathbb{R}\to\mathbb{R}$ measurable on $\mathbb{R}$?

Suppose we have a bounded differentiable function $f:\mathbb{R}\to[a,b]\subset\mathbb{R}$. Hence $f$ is continuous and measurable (in terms of standart Lebesgue measure) on $\mathbb{R}$. I want $f$ ...
1
vote
2answers
31 views

finite spectrum eigenvalue

Let $T:X \to X$ be a linear bounded operator where X is Banach space ,and $\sigma (T)$ is a finite set.Then does the spectrum consist of eigenvalues only? Any hint or counterexample is ok. thanks in ...
3
votes
1answer
20 views

Why is $C(\beta \mathbb{R})/C_0(\mathbb{R})\cong C(\beta \mathbb{R}\setminus \mathbb{R})$ as $C^*$-algebras?

Let $\beta \mathbb{R}$ be the Stone-Čech compactification of $\mathbb{R}$ (with euclidean topology) and $C_0(\mathbb{R})$ the $C^*$-algebra of continuous complex-valued functions vanishing at ...
0
votes
1answer
35 views

Exercise 1.65 of Megginson's “An Introduction to Banach Space Theory”.

Unfortunately I do not succeed in completing the following exercise: Let $X$ be a Banach space and let $T : X \to \ell^{1} (\mathbb{N})$ be a linear operator. For each $n \in \mathbb{N}$, let ...
2
votes
1answer
20 views

Weak/Weak* topologies compared to topologies generated by semi-norms from dense subset

The weak topology on normed linear space $X$ can be defined as being induced by semi-norms $\|\cdot\|_{x'}$, $x'\in X'$ with $\|x\|_{x'}=|x'(x)|$. Similarly the weak* topology is induced by ...
-1
votes
1answer
14 views

The Banach Mazur distance between n-dimensional space and $ \ell_{\infty} ^n$

Let $X$ be an $n$-dimensinal space. Is the Banach-Mazur distance $d(X,\ell_{\infty}^n)$ less than or equal to $n$? Is $d(X,\ell_{\infty}^n)$ less than or equal to some constant $C(n)$ depending only ...
0
votes
0answers
60 views

Dual space $E'$ is metrizable iff $E$ has a countable basis

I saw that it was already asked, but the book where I'm studying is slightly different. Recall some definition, if $E$ it's $\mathbb{K}$-vector space and let $\mathcal{E}$ be a vector subspace of the ...
0
votes
2answers
21 views

For a family of sets $\mathbb{U}$, $\cup_{arbitrary}(\cap_{finite} U)$ $\forall U \in \mathbb{U}$ is stable under $\cap_{finite}$.

The weak topologies of a Banach Space are constructed by taking a family $\tilde{B}$ consisting on all finite intersections of $\mathbb{U}$ and then taking arbitrary unions of sets of $\tilde{B}$. I ...
4
votes
0answers
40 views

Can we use sequences to test continuity of a weak$^*$-continuous operator?

Let $X,Y$ be Banach spaces. Now assume we have a map $T:X'\rightarrow Y'$ where $X'$ and $Y'$ are equipped with the weak$^*$ topology and not the norm topology. Can I infer from this that an operator ...
0
votes
1answer
37 views

$\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1 $ then Holder's inequality [duplicate]

If $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1 $ and $ f\in L_p $ $g\in L_q $ and $h\in L_r $ so how can I prove $$ ||fgh ||_1\le||f||_p\ ||g||_q\ ||h||_r $$
2
votes
1answer
31 views

Operator with norm

I got the following problem to solve: Let $H$ Hilbert space and $T: H \to H$ a bounded positive operator, i.e. \begin{align*} \langle x, T x \rangle \geq 0 & & \text{for all } x \in H. ...
0
votes
0answers
22 views

The precise definition of Cartesian coordinate and Euclidean space?

I'd searched them for a while, but still have not found a clear and unity definition on it. The problem really confused me. What is the precise definition of Cartesian coordinate and Euclidean space? ...
-1
votes
2answers
60 views

Proof of uniqueness about distribution in Rudin's

I'm reading Functional Analysis by Rudin, and have trouble understanding a part of the proof of theorem 6.33, in page 174. This theorem states an one-to-one relationship between a linear continuous ...
0
votes
0answers
23 views

How to prove a function is not positive definite

I have a lecture about matrix analysis. I have already know some strategies to prove that the function is positive definite. But I face difficulties when I try to see that the (bounded) function is ...
1
vote
1answer
23 views

How to show that this integral operator is bounded?

Consider the integral operator $T : C([0,1])\to C([0,1])$ given by $$Tf(t)=\int_0^1 K(t,\tau)f(\tau)d\tau.$$ I'm solving one exercise which is to show this operator is bounded. The exercise is from ...
0
votes
1answer
20 views

Isometry and Extreme points

If $X$ is a Hilbert space and either $T$ or $T^*$ is an isometry, show that T is an extreme point of the closed unit ball of $B(X)$ where $B(X)$ is bounded linear functionals on $X$. Can I get some ...
0
votes
1answer
28 views

When a set of functions becomes complete?

I know that a set of functions are said to form a complete basis on an inteval if any function on that interval can be expressed as a linear combination of the functions in the set. I also know that ...
0
votes
1answer
11 views

the sum operation in a normed space is continuous

If $ (E, \parallel \parallel) $ is a normed space , then the function $ + : E \times E \rightarrow E $ , $ ( x , y) \mapsto x + y $ is continuous. Question: to solve this exercise ... Which it is ...
1
vote
2answers
23 views

Let $f\in L^p(0,1)$ and define $f_h$

Let $f\in L^p(0,1)$ ($1\leq p<\infty$) and define $f_h$ as $$f_h(x)=\begin{cases}f(x+h)&\text{ for } x+h\in [0,1]\\ 0 &\text{ for } x+h\not\in[0,1]\end{cases}$$ Prove that for all ...
0
votes
0answers
22 views

Proof of Radon-Riesz Theorem for $L^p$ spaces

I want to know if there is a much more easy proof, in the particular case of $X=L^p, \, 1 <p <\infty$ of the following Radon-Riesz theorem: Let $X$ be uniformly convex, $(f_n)$ a sequence ...
1
vote
2answers
16 views

Application of the Krein Milman Theorem about ball $l^1$

Show that ball $l^1$ is the norm closure of the convex hull of its extreme point Define the extrem point. A point in $K$ is an extreme point of $K$ if there is no proper open line segment that ...
0
votes
3answers
100 views

Selfadjoint Operator: Basic Criterion

For symmetric operators one has: $$A\text{ symmetric}:\quad\mathcal{R}(A\pm\imath)=\mathcal{H}\implies A^*=A$$ How to prove this in an unveiling way?
3
votes
1answer
30 views

Uniqueness of endpoints of half-open line segments in linear spaces.

I try to solve the following exercise, which is Exercise 1.18 in Robert Megginson's An Introduction to Banach Space Theory. Let $X$ be a linear space, and define for any $x_1, x_2 \in X$ the line ...
1
vote
2answers
41 views

Inclusion of Schwartz space on $L^p$

I'm looking for a proof of $\mathcal{S}(\mathbb{R}) \subset L^p(\mathbb{R})$ for $1 \leq p \leq \infty$. My informal probe follow like this: For any function $f \in L^p(\mathbb{R})$ exists a ...
0
votes
1answer
15 views

Let $x,y \in l_p$, proof that $2^k (||x||^p + ||y||^p)^{2/p} \leq 2 (||x||^2 + ||y||^2)$, where $k = 2 - 2/p$ and $1<p\leq 2$

Let $x,y \in l_p$, proof that $2^k (||x||^p + ||y||^p)^{2/p} \leq 2 (||x||^2 + ||y||^2)$, where $k = 2 - 2/p$ and $1<p\leq 2$ My attempt: It's equivalent to proof the following inequality: $$ ...
0
votes
1answer
34 views

Is the sequence of functions $g_n=ng_1(nx)$ a Cauchy sequence?

Given a function $g_1(x) \in \mathcal L^2(\Bbb R)$ that satisfies: $$\int_{-\infty}^{\infty}dx \space g_1(x)=1$$ one can define a sequence of functions $g_n=ng_1(nx)$. Does $g_n(x)$ define a ...
1
vote
0answers
24 views

Integrate by parts $\int_0^\infty w' \bar w$; any nice expression for $w$ complex-valued?

Let $w$ be a complex-valued function of $t \in [0,\infty)$. At $t \to \infty$, it decays to zero. And $w_t(0)$ is prescribed. Is there any nice expression for the integral $$\int_0^\infty w' \bar w$$ ...
3
votes
1answer
30 views

What is an example of a uniformly continuous function but not absolutely continuous

Is there a function that is uniformly continuous function but not absolutely continuous. My answer is $f(x)=x^{2}, \forall x\in R$ Is this right? Are there any other?
0
votes
2answers
21 views

Number of solutions using graph

We have to find the number of solutions of $e^((-x^(2))/2)$ + $-x^2 =0$ I tried it and got one solutions by drawing graph. Is I have done correct ? My try is on :
1
vote
1answer
32 views

Exercise 1.64 of Megginson's “An Introduction to Banach Space Theory”.

Could someone verify whether my solution to the following exercise is correct? The reason I am a bit in doubt is because the chapter of which this Exercise is part consists of the Banach-Steinhaus ...
0
votes
1answer
11 views

Extending a bounded linear operator of finite rank

Let $X$ and $Y$ be normed spaces and let $W$ be a subspace of $X$. Assume that $T$ is a bounded linear operator from $W$ to $Y$, that is of finite rank. Show that $T$ can be extended to a bounded ...
0
votes
1answer
36 views

calculate Fourier Transformate

i have the following exercice: Let for all $x \in \mathbb{R},$ $f(x)= \cos x$ and $g(x)= \sin x$. Calculate $T=f \delta' + g \delta''$ for this question, i find $T=3 \delta$. Calculate the Fourier ...
2
votes
2answers
38 views

Prove that $\{g_n\}$ is a frame if and only if frame operator is continuous and continuously invertible on its range.

A sequence of distinct vectors $\{g_1,g_2,...\}$ belonging to a separable Hilbert space $H$ is said to be a Frame if there exist positive contants $A$, $B$ such that $$A\|g\|^2\leq\sum_{n=1}^\infty ...
1
vote
1answer
21 views

Lower-semicontinuity of quadratic functionals and weak convergence weirdly associated

Each random variable (rv) considered here is absolutely continuous. Let $f_{T}$ be a bounded random variable and $T>0$. Suppose we have: $$f_{T}\rightharpoonup f\tag{1}$$ weakly in probability, ...
0
votes
1answer
29 views

Isometry on Hilbert space [on hold]

If $T: H \rightarrow H$ be a function on hilbert space $H$ that satisfies $(Tx | Ty) = (x |y)$ for all $x$ and $y$. I need to show that $T$ is linear and an isometry in $B(H)$.
-1
votes
0answers
13 views

Weak-topology separates points on $X$.

Let $X$ be a normed vector space and $X^{\ast}$ be its dual consisting of all continuous functionals on $X$. I must show that weak topology separates points on X, i.e. (?) that for any $x,y \in X$ ...
1
vote
1answer
22 views

Bounded operators on inner products on Hilbert space

If we have a Hilbert space $H$ with inner product $( \cdot | \cdot)$, and let $( \cdot| \cdot)_1 $ be another inner product on $H$ such that $(x | x)_1 \leq (x | x)$ for every $x \in H$. I was trying ...
3
votes
3answers
53 views

Finding domain of $f \circ g$

I am having a small question, please don't close this before answering, I just want to know whether its a matter of convention or not. If $f(x) = \dfrac{1}{x}$ and $g(x) = \dfrac{1}{x}$ $ $ Then $f ...
0
votes
3answers
32 views

Every bounded sequence of dual space contains a subsequence which is weak* convergent

I were doing this problem in Functional Analysis of Erwin Kreyszig(part 4.9, problem 10, page 269), but got stuck in the last point to come to the conclusion. Can anyone give me some hint to move on? ...
0
votes
2answers
23 views

An exhaustion of $C_b(\Omega)$

Consider the space $\Omega=\mathbb{R}^{\mathbb{N}}$ and the space $C_b(\Omega)$ consisting of all bounded continuous functions defined in $\Omega$. Actually we are considering in $C_b(\Omega)$ the ...
0
votes
1answer
411 views

Is the derivative of a function square integrable if the function itself is square integrable in an interval?

If a function $\Phi(x)$ is square integrable in an interval $[-L, L]$, is the function $\Phi'(x)$, the derivative of $\Phi$ with respect to $x$, necessarily square integrable in the same range? In ...
1
vote
1answer
53 views

Can we embed $X'\otimes Y$ into the space of bounded, linear operators $X\to Y$?

Let $\mathbb F\in\left\{\mathbb C,\mathbb R\right\}$ $X$ and $Y$ be normed $\mathbb F$-vector spaces $X'$ denote the topological dual space of $X$ $\mathfrak L(X,Y)$ denote the space of bounded, ...