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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18 views

linear combination of compact operators not compact

Give an example of a Hilbert space, H, and a sequence of compact operators, $(S_n)_{n=1}^\infty$ on H such that i.) $\|S_n\| \leq 1$ for $n = 1,2,...$ ii.) The operators $V_N=\sum_{n=1}^N ...
2
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4answers
35 views

Why in the defn of bounded linear functional does the bound depend on $x$?

If $T : X \to Y$ is a linear functional between normed spaces, we say $T$ is bounded if $\exists M > 0$ such that $||T(x)||_{Y} \leq M ||x||_{X}$ for all $x \in X$. Usually, when we say bounded, ...
2
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1answer
140 views
+50

Two ODEs, why is one solution the solution of the other?

This question is based on Zeidler II/B, Problem 30.2. Consider the ODE: find $u:[0,T] \to \mathbb{R}^n$ s.t. $$u'(t) = F(t,u(t))$$ $$u(0) = u_0$$ given $F:[0,T]\times \mathbb{R}^n \to \mathbb{R}^n$ ...
1
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0answers
33 views

Additional assumptions on function to ensure uniform convergence

Given a sequence $u=(u_n)_{n\geq1}$ converging to $1$, I would like to prove uniform convergence of the sequence of functions $f_n$ defined by $f_n(x)=f(u_n x)$ for $x\in\mathbb{R}_+$ to the function ...
1
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2answers
118 views

Compact Operators: Trace

Given a Hilbert space $\mathcal{H}$. Consider a bounded operator: $$A:\mathcal{H}\to\mathcal{H}:\quad\|A\|<\infty$$ Regard ONB's: ...
3
votes
1answer
26 views

Compact diagonal operator

Suppose $A : H \to H$, where $H$ is a Hilbert space, is bounded. Also, $A$ is a diagonal operator with diagonal $\{a_n\}$. Show: If $A$ is compact, then $a_n \to 0$ as $n \to \infty$. Should I prove ...
0
votes
1answer
28 views

Showing that $d_\infty(x,y)$ is finite for all $x,y\in\mathscr L^\infty$

I want to show that $d_\infty(x,y)$ is finite for all $x,y\in\mathscr L^\infty$, however I am not to sure how to go about doing so. We have, $$d_\infty(x,y)=\sup_i|x_i-y_i|$$ and $\mathscr ...
2
votes
0answers
24 views

Least expensive way to “walk” with a convex potential

Let $V(x)$ be a non-decreasing, convex function on $\mathbb{R}^+$. Let $A_{L,k}$ be the space whose elements are sequences of positive integers $r= (r_i)_{i=1}^{N}$ such that $\sum\limits^N_{i=1} r_i ...
1
vote
0answers
19 views

Operators whose spectrum has a finite number of connected component

Assume that $H$ is a separable Hilbert space. Let $Q$ be the set of all operators$T \in B(H)$ such that the spectrum of $T$ has a finite number of connected component. Is $Q$ a subvector space ...
3
votes
2answers
158 views

Show that the trace class operators on a Hilbert space form an ideal

Let $(H, (\cdot, \cdot))$ be a separable Hilbert space over $\mathbb{L} = \mathbb{R}$ or $\mathbb{C}$. Suppose that $\{\phi_n\}_{n=1}^\infty$ is an orthonormal basis for $H$. Let $\mathcal{B}(H)$ ...
0
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2answers
32 views

Question about trace class operators

Let $\cal{H}$ be a Hilbert space, $T$ a bounded linear operator on $\cal{H}$, $S$ a trace class operator, then can one verify that $$|Tr(TS)|\leq\|T\|\cdot|Tr(S)|?$$
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votes
2answers
27 views

Metric spaces, manipulating the absolute value function.

I have the following problem involving the set $Y$ of infinite sequences that absolutely converge such that, $$\sum_{i=0}^\infty x_i^2 \lt\infty$$ where $x_i$ is the $i$-th term in the infinite ...
0
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0answers
15 views

Question about a proof in Clapp's paper, “On the number of positive symmetric solutions of a nonautonomous semilinear elliptic problem”

I have this: And I want to understand this proof: I don't understand the choice of $f(p)$ why it is sufficient to prove that $\lim_{p\rightarrow2^*} f(p)=f(2^*)$ since we see that ...
0
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0answers
14 views

Hyperbolic equations with time dependent coefficients associated with the time derivatives

I'm concerned with evolution equations (of second order) and am hoping for some literature hints regarding a special situation. The equations I'm working with basically look like (complemented with ...
0
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1answer
10 views

Approximation by $\mbox{Im }(t-z)^{-1}$ with $\mbox{Im } z > \epsilon$

It is a standard fact of harmonic analysis that the span of the functions $$g_z(t) = \mbox{Im } (t-z)^{-1},$$ ranging over all $z \in \mathbb{C}$ with $\mbox{Im } z > 0$, is dense in ...
1
vote
1answer
38 views

Are infinite-dimensional singletons measurable?

Consider the wiener measure space $C[a,b]$ of all real-valued continuous functions on $[a,b]$ with the wiener measure $\mu$ on the $\sigma$-algebra $\mathcal{A}$ of Carathéodory measurable sets in ...
0
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1answer
37 views

Is there an error in the solution for this exercise?

I have this exercise: H is a complex hilbert space. And T is a compact operator on H. Show that if H is not separable, then 0 is an eigenvalue of T. Hint: Use lemma 1, and theorem 2. The ...
0
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1answer
15 views

The relationship between function space embeddings and their respective inequalities

Let $L^{p,\infty}$ be the weak $L^p$ space consisting of measurable functions $f$ satisfying \begin{equation*} ||f||_{p,\infty}:=\sup_{\rho}\rho\lambda (|f|>\rho)^{\frac{1}{p}}<\infty . ...
4
votes
1answer
365 views

Zeros/poles at Laplace and at Fourier Transform

I recently started "relearning" the Laplace transform, and I noticed something. It seems to me that the intuitive idea of poles and zeros is different between these two transforms! For example, in ...
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0answers
17 views

Equivalence of the Sobolev norm $\| f\|_{W^{k,\infty}}$ and $\| f\|_{H^{k,\infty}}$.

I want to know that when $k=0,1,2,...$, the Sobolev norm $$ \| f\|_{W^{k,p}} :=( \sum_{|\alpha|\le k} \|\partial^\alpha f\|_{L^p(\mathbb R^n)}^p )^{1/p} $$ is equivalent to the norm $$ \|f\|_{H^{k,p}} ...
3
votes
1answer
25 views

Manipulating the maximum function, metric spaces.

I am trying to show that the supremum metric, $d_{\infty}$, is indeed a metric on $\mathbb R^2$. I have shown that the first two properties of a metric space hold, but am having trouble showing the ...
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0answers
46 views
+50

How to construct examples of functions in the Spaces of type $\mathcal{S}$

We know that there are $3$ types of $\mathcal{S}$-type Spaces, namely $\mathcal{S}_\alpha\:,\: \mathcal{S}^\beta\:,\:\mathcal{S}_\alpha^\beta$. $\mathcal{S}_\alpha: |x^k\varphi^{(q)}(x)|\le ...
5
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2answers
57 views

What is the implication that $\| \cdot \|_2$ and $\| \cdot \|_\infty$ are equivalent norms on $\mathbb{R^2}$

Given $\mathbb{X}$ = $\mathbb{R^2}$, consider $\| \cdot \|_2$ and $\| \cdot \|_\infty$ We can show that $\| x \|_\infty \leq \| x \|_2 \leq \sqrt2 \| x \|_\infty$ Hence $\| \cdot \|_2$ and $\| ...
0
votes
0answers
12 views

A pde that cannot solves by Lax-Milgram theorem

Consider the following pde: $-u''(x)+au'(x)+bu(x)=f(x) \qquad\text{in}\; (0,1)\\$ $u'(0)=\alpha\\$ $u'(1)+u(1)=\beta$ How could I prove that it has a nontrivial solution? The bilinear form ...
1
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0answers
14 views

Inequality in a Dirichlet BVP

For the Drichlet boundary value problem $Lu = -u''+p(x)u'+q(x)u = f(x), \; \; \; x \in I=[a,b]$. with $u(a)=u(b)=0$. Then for $v \in H^2(I) \cap H_0^1(I)$ show that $\left\lvert \right\rvert v ...
0
votes
1answer
302 views

Positive unbounded operators

Let $T$ be an operator in $H$. We say self adjoint $T$ is positive iff $(\forall x\in H)\langle Tx,x\rangle \geq 0 $. As in the case of bounded operators, it is true that a self-adjoint operator $T$ ...
6
votes
1answer
51 views

Is this set of random variables a Hilbert space?

Consider a sequence of i.i.d. random variables $\left\{ {{\varepsilon _t}} \right\}_{t = 1}^\infty $ with $E\left( {{\varepsilon _t}} \right) = 0$ and $E\left( {\varepsilon _t^2} \right) = {\sigma ...
2
votes
1answer
24 views

Big an open ball inside small open ball in metric space

When i was reading a book "Elements of the Theory of Functions and Functional Analysis " ([A. N. Kolmogorov, S. V. Fomin) I encountered with very interesting(for me) problem. Problem: Creat a ...
9
votes
2answers
75 views

Existence of a function

I need some help: I am thinking about this problem. Any advice would be appreciated. Let's fix $\epsilon>0$. Does there exists some $f\in C^0([0,\pi])$ such that: ...
3
votes
1answer
46 views

Independent symmetric 3-valued random variables in $L_p$

Consider the following excerpt from this paper: Given $1<p<2$, $0<w\leq 1$ and $n\in\mathbb{N}$, we fix once and for all a sequence $f_j^{(n)}=f_{p,w,j}^{(n)}$, $1\leq j\leq n$, of ...
0
votes
1answer
68 views

Ordering: Compactness

Given a Hilbert space $\mathcal{H}$. Denote selfadjoints: $$\mathcal{S}(\mathcal{H}):=\{A\in\mathcal{B}(\mathcal{H}):A=A^*\}$$ Introduce an order: $$A\leq A':\iff\sigma(A'-A)\geq0$$ Denote ...
0
votes
1answer
22 views

Finite Rank Operator: Continuity

I keep forgetting it, so... Given Banach spaces $X$ and $Y$. Then it is wrong: $$\dim\mathcal{R}F<\infty:\quad F\in\mathcal{L}(X,Y)\implies F\in\mathcal{C}(X,Y)$$ Can I construct such?
0
votes
1answer
31 views

How to compute high order differential?

Let $f: E \to \mathbb{R}$ sending $x \mapsto \|x\|$ and make some simple hypothesis $E$ is a Hilbert Space Let's say that the norm $\|\cdot\|$ is derived from a scalar product So we can easily ...
0
votes
1answer
34 views

Sobolev space for Mixed Dirichlet - Neumann boundary condition

Consider the subset $\Omega\subset \mathbb{R}^N$ with boundary $\partial\Omega$ sufficiently regular and let $\Gamma\subset\partial\Omega$ be a $(N-1)$- dimensional submanifold of $\partial\Omega$. ...
4
votes
0answers
82 views

Random variables that span copies of $\ell_p$

Consider the coin-toss measure $\mu$ on $\{0,1\}^\mathbb{N}$. Within this framework it is easy to construct a sequence of independent, symmetric Bernoulli random variables. Indeed the point-evaluation ...
2
votes
1answer
79 views

Difficult to read about different subjects simultaneously, should I leave one for now? [on hold]

I learn math by reading books. Usually I read 3 books (about 3 different subjects) simultaneously and switch focus every couple of days. The books i'm studying right now are Rudin's functional ...
2
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1answer
40 views

Calculus stay to Real Analysis as $x$ stay to Functional Analysis

Hi guys i had a look to book which treat the subject of Calculus (of course...) Analysis and Functional Analysis. Is that correct to state that Calculus is more focused on "computing" while ...
2
votes
2answers
47 views

Second differential of the norm in an infinite dimensional Hilbert space

Let $f: E \to \mathbb{R}$ sending $x \mapsto \|x\|$ and make some simple hypothesis $E$ is a Hilbert Space Let's say that the norm $\|\cdot\|$ is derived from a scalar product [solved] So we can ...
2
votes
1answer
43 views

Equivalent norms and inner product

It is not hard to give examples of normed spaces which are not inner product spaces. Now let $(V, \|\cdot\|)$ be a normed space. Is it always possible to construct an inner product on $V$ which gives ...
6
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1answer
39 views

structure theorem for Banach spaces

The following is a theorem in the Banach Algebra Techniques in Operator Theory by Douglas: Here are my questions: Could one come up with some reference (or proof) regarding the remark right ...
1
vote
1answer
37 views

Can we say $\| f\|_{\dot H^s(\mathbb R^n)} \le \| f \|_{\dot H^q(\mathbb R^n)}$ if $s\le q$?

If $s\le q$, then can we say that $$ \| f\|_{\dot H^s(\mathbb R^n)} \le \| f \|_{\dot H^q(\mathbb R^n)} $$ holds? Here the homogeneous Sobolev seminorm $\|f\|_{\dot H^s(\mathbb R^n)} = ...
2
votes
0answers
24 views

derivative of a linear operator

I am a little confused by the proof in the picture. Doesn't the calculation show $dB(u)h = B(h_1,u_2)+ B(u_1, h_2)$?
2
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1answer
31 views

Symmetric function of two normal distribution implies bilinear

This question is related to my previous question which was partially answered my @MichaelHardy. Let $X$ and $Y$ be two independent standard normal random variables. Now, suppose that ...
0
votes
1answer
53 views

Topology on compactly supported smooth functions

I'm confused by a set of lecture notes I'm reading and would like help in understanding what's going on. First, there is the following nice theorem. Theorem. The topology of a locally convex space is ...
4
votes
1answer
23 views

“Occupation time” nonlinear functional measurable?

My question is for which functions $f$ the following nonlinear functional $f\rightarrow\int \mathbf{1}_B(f(x))dx$ is Borel measurable; $B\in\mathcal{B}(\mathbb{R})$ and $\mathbf{1}_B(.)$ is a ...
3
votes
0answers
37 views

looking for some exercise to test understanding of covector

I am trying to understand the concept of "covector" so that I can work with it for problems I come across in PDE. My knowledge on differential geometry is very little. And my topological background ...
1
vote
2answers
37 views

Trace evaluation via complex analysis

We are given $U$, $V$ unitary matrices of size $N \times N$ whose spectral decomposition is known (in my specific problem, $N=4$, and $U$, $V$ are matrices with real coefficients but we can keep it ...
1
vote
1answer
32 views

How can I prove Dini's theorem using the Baire Category theorem?

Let $(X,d)$ be a compact metric space. For each $n \in \mathbb{N}$ we have $ f_n:X \to \mathbb{R}$ be a continuous function such that $f_n(x) \geq0 \forall x \in X$ . Assume that for all $x \in X$ the ...
4
votes
1answer
52 views

Some questions about an exercise about $C^\infty \subset L^\infty$

Let $$ L^\infty (\mathbb R) = \{f : \mathbb R \to \mathbb C\mid \text{essential sup of } f < \infty \text{ and } f \text{ Borel measurable} \}$$ and $$ C^\infty (\mathbb R ) = \{ f: \mathbb R ...
1
vote
1answer
27 views

Is there a name for embedding of Lebesgue spaces $L^q(a,b) ⊂ L^p(a,b)$

Let $1 ≤ p ≤ q ≤ ∞$ then $L^q(a,b) ⊂ L^p(a,b)$ where $L^q(a,b), L^p(a,b)$ are Lebesgue spaces Is there a name for this relationship? Is this the Sobolev embedding theorem?