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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4
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9 views

Brezis Exercise 6.10, why does it follow that $Q(S) \in \mathcal{K}(E)$?

Here is the third part of Brezis, Exercise 6.10. Let $Q(t) = \sum_{k = 1}^p a_kt^k$ be a polynomial such that $Q(1) \neq 0$. Let $E$ be a Banach space, and let $T \in \mathcal{L}(E)$. Assume that ...
0
votes
0answers
9 views

Provide example for a certain reaarangement invariant Banach space of Lebesque-measurable functions

I look for an example of a rearrangement invariant Banach space X of Lebesque-measurable functions on $(0,1)$, preferably, which meets the following criterion: There exists a function $g>0$ in $X$ ...
4
votes
1answer
22 views

$E$, $F$ two Banach spaces, does every operator $T \in \mathcal{L}(E, F)$ satisfy a certain property?

Let $E$ and $F$ be two Banach spaces, and let $T \in \mathcal{L}(E, F)$. Consider the following property (P). For every weakly convergent sequence $(u_n)$ in $E$, $u_n \rightharpoonup u$, then ...
0
votes
1answer
19 views

Proving a function defined in terms of a $C_{0}$-semigroup is continuously differentiable

Suppose that $u\in C([0,\infty))\cap C^{1}([0,\infty))$ is a solution of $$\begin{cases}u'(t)=Lu(t),& t\ge 0, \\ u(0)=x\end{cases}$$ Fix $t>0$ and define the function $$v(s):=T(t-s)u(s),\qquad ...
1
vote
1answer
23 views

Distance in metric space, triangle inequality problem

Let $(X, d)$ be a metric space. Let $t\in (0,1]$. Show that $d^t: X\times X\to\mathbb{R}$ $$d^t (x,y) := d(x,y)^t, \forall x,y\in X$$ is also a distance function. Problematic bit is the triangle ...
1
vote
0answers
14 views

Can someone provide some examples to illustrate the difference between Pointwise equicontinuity and Uniform equicontinuity?

I don't know what is with the subject of pointwise and uniform equicontinuity, pretty much all the material you can find online are either: Proofs i.e. pointwise equicontinuity is uniform ...
0
votes
1answer
15 views

Parseval relation on inner product space for $\langle x,y \rangle$

Exercise 3.6-4 in Kreyszig asks to show that $\langle x,y \rangle = \sum_k \langle x,e_k \rangle \overline{\langle y,e_k \rangle}$ using the "Parseval relation": $\sum_k |\langle x, e_k \rangle |^2 = ...
2
votes
1answer
46 views

Lipschitz map between metric and normed spaces

Let be $F:(X,d)\to V$ a map between $(X,d)$ metric space and $V$ normed space, such that for each $f\in V'$ (linear and continuous), $f\circ F$ is lipschitz map. Show that $F$ is a Lipschitz map. I ...
0
votes
1answer
16 views

Every vector in a Hilbert space has a Fourier representation wrt an orthonormal sequences?

I'm reading Kreyszig's text, and there is a Theorem in section 3.5 stating: Theorem: Let $(e_k)$ be an orthonormal sequence in a Hilbert space $H$. Then 1) If $\sum_{k=1}^\infty \alpha_k e_k$ ...
-1
votes
0answers
20 views

What does a subscripted norm mean in context of functional spaces? [on hold]

While studying fundamentals of Finite Elements, I encounter this notation very frequently. $$\|u\|_{H}$$ I understand that it is a norm, but what does the subscript mean? Does it have to do ...
0
votes
0answers
7 views

Convergence of product of functions

Let say we have a sequence of functions $(f_ng)$ in $L^2[a,b]$, where $g$ is in $L^2[a,b]$, that converges to some function $h\in L^2[a,b]$. i.e. $f_ng\to h$ in $L^2$ as $n\to\infty$. ($f_ng$ ...
3
votes
1answer
21 views

Infinite dimensional topological vectorspaces with dense finite dimensional subspaces

Consider $\mathbb R$ as a $\mathbb Q$ vector space. Using the usual metric on $\mathbb R$, we find: $\mathbb Q \subset \mathbb R$ is dense and one dimensional (indeed every non-zero subspace appears ...
-3
votes
0answers
10 views

> example of non supperadditve function [on hold]

I need an example: 1.l-superadditive 2.monotono 3.non superadditive. please help me!
2
votes
1answer
27 views

Dual map is zero if and only if map is zero

A problem from Linear Algebra Done Right (Third Ed): Suppose $W$ is finite dimensional and $T \in \mathcal{L}(V, W)$. Prove that $T=0$ if and only if its dual $T' \in \mathcal{L}(W', V')=0$. I am ...
1
vote
1answer
22 views

Computation of an integral depending on the Legendre polynomials

Let $P_l$ be a Legendre polynomial ($l$ is an integer). I want to know why the quantity $$ v_l(k):=(-i)^l\int_{-1}^{+1}\mathrm{e}^{ikx}\,P_l(x)\;\mathrm{d}x $$ is real?
0
votes
2answers
31 views

Completeness and orthogonal projection

a. Which are the properties that define an orthogonal projection? Give a precise definition. b. What does completeness mean? Please state both the definition and an example (without proof) ...
1
vote
1answer
22 views

Find an operator on $C[0,1]$ with a given compact set in $C$,the complex field

Let $K$ be a non-empty compact subset of $C$,the complex field. Does there exist an operator in $\mathscr{B}(C[0,1])$ such that $\sigma(A)=K$? $A$ is a multiplication operator iff $K$ is non-empty ...
0
votes
0answers
16 views

Symbol for continuous dual

Let $X$ be a vector space. The dual is the set of all linear forms from $X$ to $\mathbb{F}$.(can be $\mathbb{R}$ or $\mathbb{C}$) That is, $X^* = \operatorname{Hom}(X,\mathbb{F})$. If $X$ is also ...
0
votes
0answers
12 views

Existence of nuclear dominating positive definite kernel

Let $\mathcal{X}$ be a metric space and $k: \mathcal{X} \times \mathcal{X} \to \mathbb{R}$ a continuous positive definite kernel. Can we always find a positive definite kernel $r$ such that $r \gg k$ ...
2
votes
2answers
30 views

Is a self-adjoint operator continuous on its domain?

Let $H$ be a Hilbert space, and $A : D(A) \subset H \rightarrow H$ be an unbounded linear operator, with a domain $D(A)$ being dense in H. We assume that $A$ is self-adjoint, that is $A^*=A$. Since ...
1
vote
0answers
22 views

The convergence rate of the derivative of a sequence of function

Let $v_\delta$ be a sequence of continuous diff'able function on $(-1,1)$ and $0\leq v_\delta\leq 1$. For each $\delta>0$, assume that $v_\delta(\delta)=v_\delta(-\delta)=1$ and $v_\delta(0)=0$. We ...
0
votes
0answers
13 views

An example of convergence to Young measures

$\newcommand{\R}{\mathbb{R}}$ $\newcommand{\lam}{\lambda}$ I am trying to prove the following claim: Let $\{u:[0,1]\to \mathbb{R} \mid u \, \, \text{ is differentiable a.e}, u(0)=u(1)=0 \}^{*} $. ...
0
votes
1answer
33 views

Why isn't every Hamel basis a Schauder basis?

I seem to have tripped on the common Hamel/Schauder confusion. If $X$ is any vector space (not necessarily finite dimension) and $B$ is a linearly independent subset that spans $X$, then $B$ is a ...
6
votes
1answer
44 views

Differentiation as Rotation

I am trying to make a connection between linear algebra and the Fourier transform. Functions form a vector space and differentiation is an operator. Fourier transforming a function from what i ...
0
votes
1answer
45 views

Show that $|T(x) - T(y)| \lt |x-y|$ when $x \neq y$ but the mapping has no fixed points.

Consider $X = \{ x \in \mathbb{R} \mid 1 \le x \lt \infty \}$, taken with the usual metric of the real line, and $T \colon X \to X$ defined by $x \mapsto x +x^{-1}$. Show that $|T(x) - T(y)| \lt ...
0
votes
1answer
27 views

Triangle inequality with a twist

Assume $t>0$ and $x,y,z\in [0,t)$ how would one go about showing $$\min \{|x-y|,t-|x-y|\}\leq\min \{|x-z|,t-|x-z|\}+\min \{|z-y|,t-|z-y|\} $$ If the first one materializes from every minimum, then ...
1
vote
1answer
31 views

How to construct a complete set in $L^2(\mathbb{R}^3)$ starting with the Spherical Harmonics?

The Spherical Harmonics form a complete set of functions on the sphere $S^2$, so that any function of $f: S^2\to \mathbb{R}$ can be written uniquely as $$f(\theta,\phi)=\sum_{l=0}^\infty ...
0
votes
1answer
19 views

How can the inverse of an operator between Hilbert spaces H,K be defined on the dual of H?

I need some help to understand the following statement. Let $A$ be an operator defined as follows: $Av = -\Delta v - \nabla \text{div} u$ It is known that the operator $A$ is positive self-adjoint ...
0
votes
1answer
25 views

Doesn't this $L^p$ norm estimate for all $p$ give me an $L^\infty$ bound?

Let $r_n \to \infty$ as $n \to \infty$. We have that $$\lVert v \rVert_{L^{r_n}(\Omega)} \leq C\lVert v \rVert_{L^{r_0}(\Omega)} < \infty$$ for all $n$, where $C$ is independent of $v$ and ...
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votes
0answers
15 views

Prove that the standard orthonormal sequence $(e_n)^\infty_{1}$ is complete in $l^2$. [on hold]

Prove that the standard orthogonal sequence $(e_n)^{\infty}_{1}$ is complete in $l^{2}$. Where $(e_{n})$ is the sequence with nth component equal to 1 and all others zero.
0
votes
0answers
12 views

Prove that $\left\|f_n(x)-g_n(x)\right\|^2 = \|f_n(x)\|^2+ \|g_n(x) \|^2-2\operatorname{Re}\int_{\mathbb R} f_n(x)\overline{g_m(x)} dx$

Let $\{f_n(x)\}_{n\in\mathbb Z}$, $\{g_n(x)\}_{n\in\mathbb Z}$ be two sequence of square-integrable functions: $f_n, g_n\in L^2(\mathbb R)$. Prove that $$\left\|f_n(x)-g_n(x)\right\|^2_{L^2(\mathbb ...
1
vote
1answer
18 views

Dissipativity for Hilbert spaces

I want to prove that an operator $A:D(A)\to X$ is dissipative $\iff$ $\text{Re}\langle Ax,x\rangle\le 0$ $\forall x\in D(A)$. The proof for this is actually sketched on the Wikipedia page for ...
-1
votes
1answer
32 views

Is every topological space is measurable?

Actually I am learning about measure theory. But I have confusion between topological space and measurable . Is there any relationship among them or not?
0
votes
0answers
13 views

prove that a non constant linear functional from a normed linear space X is discontinuous if and only if Z(f) = {x in X | f(x) = 0} is dense in X. [duplicate]

I could prove that if Z(f) is dense in X then the functional must me discontinuous but I am not able to prove the other way round
0
votes
1answer
13 views

The w*-extension of a bounded linear functional

Let Y be a Banach space and assume that $X$ is a $w^*$-closed subspace of $Y^*$. Let $f$ be a bounded linear functional on $X$. Does there exist any $w^*$-continuous linear functional $\phi$ on $Y^*$ ...
1
vote
1answer
18 views

An example of smooth compactly supported function with non-vanishing Fourier transform

I am trying to find an explicit example of smooth real-valued compactly supported function $u$ in $\mathbb R^n$, $n \geq 2$, such that its Fourier transform $\widehat u$ does not have zeros. By the ...
1
vote
0answers
39 views

Fourier Series in Functional analysis [on hold]

Would you please solve this question? I really have problem with this kind of questions.
2
votes
0answers
33 views

How to view Stone-Cech compactification of the real line?

I am going through Arveson's A Short Course on Spectral Theory and have come across an exercise constructing $\beta\mathbb{R}$ using the Gelfand map. I was wondering if there is an explicit ...
0
votes
1answer
31 views

Is $W^{1,2}_0$ a Hilbert space?

I came across the term $W^{1,2}_0$. Just a quick question on what the 0 means and is it a Hilbert space? I know $W^{1,2}$ is a Hilbert space. Thanks!
4
votes
0answers
47 views

On the definition of commuting self adjoint operators.

I'm reading Mathematical Methods in Quantum Mechanics by Gerald Teschl and I came across the following exercise whose statement is causing me some troubles. It goes like this: Let $A$ and $B$ two ...
0
votes
0answers
18 views

Showing coercivity of the bilinear form associated with a robin boundary value problem

I'm trying to show the existence and uniqueness of weak solutions to the following boundary value problem: \begin{align} -\nabla \cdot ( k \nabla u) &= f \quad \text{in } \Omega \subset ...
1
vote
1answer
21 views

Find the strong form of a PDE from the weak form.

I'm having a little difficulty understanding how to find the strong form of a PDE given the weak form. For example, I have the weak form as: $\displaystyle\int_\Omega [a(x)\nabla u\cdot\nabla ...
1
vote
2answers
55 views

Why to introduce norms of vectors?

I am studing Euclidean, metric and normed spaces. What I don't get it is why should I norm a vector. It is usually squared? Why should it be always positive? I've asked this to many people and nobody ...
0
votes
0answers
21 views

Show $A=\{x\in \Bbb{R}^n|\sum_{j=1}^{n}|x_j|^p\le 1\}$ is Jordan measurable for $p>0$

Show $A=\{x\in \Bbb{R}^n|\sum_{j=1}^{n}|x_j|^p\le 1\}$ is Jordan measurable if $p>0$. I did show it is a bounded set because if there exists $x^{(N)}\subset A $ such that $||x^{(N)}||\to \infty $ ...
0
votes
1answer
28 views

If $\|\cdot\|_{1}\le\|\cdot\|_{2}$ then $\|\cdot\|_{2}\le M\|\cdot\|_{1}$ [duplicate]

Let $(X,\|\cdot\|_{1})$ and $(X,\|\cdot\|_{2})$ be complete normed vector spaces and $\|x\|_{1}\le\|x\|_{2}$ $\forall x\in X$. I want to prove that $\exists M>0$ such that $\|x\|_{2}\le ...
4
votes
5answers
63 views

About inner products, norms and metrics

Do these three kinds of vector spaces, those with an inner-product, those with a norm and those with a metric, are the same sets of vector spaces? At least for finite dimensional vector spaces all of ...
1
vote
1answer
66 views

Every Hilbert space is isometrically isomorphic with $\ell^2$

Let $H$ be a hilbert space and let $\{u_\alpha\}_{\alpha \in A}$ be a orthornormal basis ($A$ is not supposed to be countable a priori). Then there is an isometric isomorphism between $H$ and ...
1
vote
1answer
37 views

How to define properly the radial and angular dependence of a function?

Recently I've came across the following situation when studying Quantum Mechanics: suppose we have two operators $A,B$ on the space of functions $L^2(\mathbb{R}^3)$ and suppose they have the following ...
0
votes
1answer
30 views

How does the general spectral theorem generalize the simpler versions?

I read the following version of the spectral theorem in Banach Algebra Techniques in Operator Theory by Douglas: I'm trying to understand why this is a generalization of the following version, ...
2
votes
2answers
53 views

Why can entire function be written as exponential, and why is it bounded in this way?

Let $A$ be a commutative complex Banach algebra with unit element $e$. Suppose now that $f(x) \in \sigma(x)$ for every $x \in A$ where $\sigma(x)$ denotes the spectrum of $x$. Now, let $x\in A$ and ...