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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10 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 ...
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
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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 ...
0
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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 $X$ ...
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 ...
0
votes
1answer
16 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 = ...
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
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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 ...
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
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) ...
0
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1answer
46 views

What is the name of the following partial differential operator?

What is the name of the following partial differential operator? $$\sum_{|\alpha| \leq n} a_\alpha (\frac{\partial}{\partial x})^\alpha$$ Thank you!
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 ...
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
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 ...
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 ...
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 ...
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 $n\to\infty$. ($f_ng$ ...
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 ...
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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!
1
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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
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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 \}^{*} $. ...
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
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 ...
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 ...
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
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 ...
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
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 ...
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 ...
14
votes
1answer
317 views

Mathematical meaning of certain integrals in physics

While studying on texts of physics I notice that differentiation under the integral sign is usually introduced without any comment on the conditions permitting to do so. In that case, I take care of ...
-2
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 ...
2
votes
1answer
93 views

$\nabla^2(\|\boldsymbol{x}-\boldsymbol{x}_0\|^{-1})=-4\pi\delta(\boldsymbol{x}-\boldsymbol{x}_0)$ with distributions defined on Schwartz space

I know, from a recent enlightening answers received here, that, if we define the distribution represented by Dirac's $\delta$ on the space $K$ of test functions of class $C^\infty$ whose support is ...
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 $Y^*$ ...
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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?
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0answers
13 views
0
votes
1answer
29 views

Compact operators on $L_p$

Consider $M_n(\mathbb{C})$ as $B(\ell_p^n)$ for $n\in\mathbb{N}$ where $p\in[1,\infty)$, and include $M_n(\mathbb{C})$ in $M_{n+1}(\mathbb{C})$ as the upper left corner. Is it true that ...
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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 ...
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 ...
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
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!
3
votes
2answers
42 views

Divergence of squared sum of Chebyshev Polynomials $\equiv L+R$ has empty point spectrum

The Chebyshev Polynomials of the second kind $U_n$ are the solutions of the differential equation $$(1-x^2)U_n''(x)-3xU_n'(x)+n(n+2)U_n(x)=0$$ Alternatively they are defined inductively: $$U_0(x)=1 ...
4
votes
1answer
73 views

If a linear operator preserves positive functions, then it leaves some linear functional invariant

I have the following question which I cannot seem to make any progress on: Suppose that $T:C[0,1]\to C[0,1]$ is a linear operator satisfying that $Tf\geq 0$ whenever $f\geq 0$, and $T1=1$. Show that ...
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 ...
0
votes
1answer
31 views

distance in Hilbert space

W is closed and convex subset of Hilbert space $H$. Suppose $x\in H$, $x_0\in W$. I am trying to proof, that $d(x, W) = ||x-x_0||$ iff $\Re(<x-x_0, y-x_0>)\le 0$ for all $y\in W$. Could anyone ...