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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Reference For a PDE text that treats non homogenuous boundary conditions Rigorously

I am interested in reading a text or paper where elliptic and parabolic PDE's are discussed on bounded domains with non-homogeneous boundary conditions. I haven't been able to find anything in the ...
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7 views

Behavior of norm on matrix algebra under multiplying matrix of scalars

If $A$ is a Banach algebra and I equip $M_n(A)$ with the norm $\|[a_{ij}]\| = \max_i\sum_j \|a_{ij}\|$, do I have $\|Z[a_{ij}]\|\leq\|Z\| \|[a_{ij}]\|$ when $Z$ is a matrix with complex scalar entries ...
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2answers
45 views

Please give me an example $d:C[\mathbb{R}]\times‎ C[\mathbb{R}]\longrightarrow\mathbb{R}$ Such that: [on hold]

I need an example $d$ such that: $$d:C[\mathbb{R}]\times‎ C[\mathbb{R}]\longrightarrow\mathbb{R}$$ $$C[\mathbb{R}]=\lbrace f:\mathbb{R}\longrightarrow\mathbb{R}\ | \ f ‎‎\textit{Is differentiable on ...
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1answer
18 views

Question on complete spaces, longer, more specific question.

Let $S \subset C^2[0,1]$ (set of two times differentiable functions $f(x)$ on $[0,1]$) which satisfy the following: $$\int_0^1 f(x)\,dx\leq3$$ Question is $(S,d)$ is a complete metric space, ...
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2answers
25 views

Prove that the normed spaces $(C[0,1], \| \cdot\|_2)$ and $(C[a,b], \|\cdot \|_2)$ where $\| \cdot\|_2$ is the Euclidean norm are isometric.

Essentially I'm looking for a bijection $f: C[0,1]\to C[a,b]$ such that$$\|f(x) \|_2=\| x\|_2$$ I don't know how to go about finding this function, but I do know that it is possible. $$\| x \|_2 = ...
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0answers
13 views

Fraction of Lipschitz functions among absolutely continuous ones

Is it true that the space of Lipschitz functions on $S^1$ is a $G_\delta$ subset of the space of absolutely continuous functions on $S^1$? In which topologies ($L^p$, uniform, $C^k$, etc) it is true? ...
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2answers
14 views

Radial Limits of Singular Inner Functions

Given a positive singular measure $\mu$ on $[-\pi,\pi]$, we define a singular inner function by $$S(z)=\exp\left(-\int_{-\pi}^{\pi}\frac{e^{i\theta}+z}{e^{i\theta}-z}\,d\mu(\theta)\right).$$ It is ...
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0answers
14 views

Closedness of the range of differential operator first order

The fact that the range of gradient from $H_0^1$ to $L_2$ is closed is well known. In general we can define some kind of weak derivative in the form \begin{equation} Du=\sum_{i,j}a_{ij}\partial_i u_j ...
1
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1answer
17 views

Continuous Sobolev Embedding

Does Sobolev spaces $H^s$ continuously embed into $L^2$? It seems like this is the case from this post https://en.wikipedia.org/wiki/Rigged_Hilbert_space where can i find a list of continuous ...
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0answers
22 views
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1answer
38 views

Compactness of operators [on hold]

Let $X$ be a separable Hilbert space. Let $A\in\mathcal{L}\left(X\right)$, a bounded linear operator on $X$, which is compact. Let $B$ be an operator in $X$, boundedly invertible, that is ...
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1answer
42 views

$T:X \to Y$ bounded linear map and $X$ separable implies $Y$ is separable?

Let $T:X \to Y$ be a bounded linear map between Banach spaces. Suppose that $X$ is separable. Is it true that $Y$ has to be separable? I think yes, since the map is continuous it takes the ...
2
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1answer
25 views

Lifting idempotents from a quotient of a Banach algebra

In a quotient of a Banach algebra $A$, if an invertible element is connected to the identity by a continuous path of invertibles, then it can be lifted to an invertible element in $A$. Is there an ...
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0answers
20 views

Quasi-Banach algebras

We know that the space $Lp(0, 1)$, when $0<p<1$ is quasi-Banach spaces and has a trivial dual; $L_{p}(0,1)^{*}=\{0\}. $But its not algebra. Is there any quasi-Banach algebra with trivial dual? ...
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0answers
22 views

Stochastic process, Fourier transform and $L^2$

Consider a time domain signal received at a sensor $x(t)$ over some time $t$, and we have performed Fourier transform on $x(t)$ to obtain $X(w)$. While performing Fourier Transform to find the ...
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0answers
18 views

Evans PDE Chapt 5 problem 4, existenc of smooth functions form a partition of unity

I have to be honest that I am very lost on the same kind of problem about proving existence of smooth function. I have not done much topology, hence unfamiliar with the "covering" business. In the ...
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0answers
32 views

Positive definite function and covariance matrix.

I tend to view positive definite function as a function of elements of positive definite matrix. A reference is: https://en.wikipedia.org/wiki/Positive-definite_function My question in essence: is ...
3
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0answers
47 views

How to prove this is a self-adjoint operator?

I have this operator from $H^1_0$ to $H^1_0$ defined by: $$Au(t)=\int_0^1 G(t,s) f(s,u(s))\mathsf ds$$ where $$G(t,s)=\begin{cases} t(1-s), &t\leq s\\s(1-t), &s\leq t.\end{cases}$$ I want to ...
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1answer
23 views

If a sequence of self-adjoint linear operators is convergent, show that its limit is self-adjoint.

If a sequence of self-adjoint linear operators is convergent, show that its limit is self-adjoint. I'm lost on this problem. I don't know how to even start this. Any solutions or hints would be ...
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0answers
29 views

Calculating the local minimum of a function

Regarding this function $f(x,y)=1007x^2-x^{2014}+(e^y-1+2x^2)^2$. I want to find the strict local minimum of $f$. I started calculating $\nabla$: $\nabla (1007x^2-x^{2014}+(e^y-1+2x^2)^2)$ ...
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0answers
11 views

Wave Operators: Adjoint

Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}H_\#\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote their evolutions: ...
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1answer
34 views

On the space $l_2$ we define an operator $T$ by $Tx=(x_1, {x_2\over2}, {x_3\over3}, . . . )$. Show that $T$ is bounded, and find its adjoint. [duplicate]

On the space $l_2$ we define an operator $T$ by $Tx=(x_1, {x_2\over2}, {x_3\over3}, . . . )$. Show that $T$ is bounded I know that $||T||\leq 1$, but I don't know how to show this. Any solutions or ...
3
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2answers
44 views

Projections: Ordering

Given a unital C*-algebra $1\in\mathcal{A}$. Consider projections: $$P^2=P=P^*\quad Q^2=Q=Q^*$$ Order them by: $$A=A^*:\quad A\geq0:\iff\sigma(A)\geq0$$ Then one has: $$P\leq Q:\quad P=PQ=QP$$ How ...
3
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3answers
46 views

Definition in Lax “sequence of continuous functions tending to $\delta$”, are distributions needed for understanding?

I'm trying to read Lax's functional analysis. In chapter 11 he makes a definition which I don't like. A sequence of continuous functions ${k_n}$ on a $[-1,1]$ tends to $\delta$ if $\int_{-1}^{1} ...
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0answers
17 views

A question about passing to limits in Banach lattices

I would be grateful if one could confirm that the following argumentation is fine. Suppose that $L$ is a Banach lattice and $(L_n)$ is an increasing sequence of sublattices of $L$. Given two positive ...
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0answers
19 views

question about $L^2([0,T]\times \Omega)$ and iterated integral [duplicate]

Let $X=[0,T]\times \Omega$ where $\Omega$ a bounded domain. Consider the space $L^2(X)$, so $u \in L^2(X)$ if $$\int_{[0,T]\times\Omega}|u|^2 < \infty.$$ Is it true that the integral $$\int_0^T ...
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1answer
36 views

Is the set defined by inequality $\|Tx\|^2\leq\|T^2x\|\|x\|$ a subspace of a Banach space?

Let $X$ be a complex Banach space and $T$ be a bounded linear operator on $X$. Put $Y=\{x\in X:\|Tx\|^2\leq\|T^2x\|\|x\|\}$. Is $Y$ a subspace of $X$? I know is that $Y$ is closed and $aY$ is ...
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0answers
11 views

Wave Operators: Cook

Problem Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}H_\#\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote for shorthand: ...
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3answers
41 views

Show that the operator sequence $ A_n = 1/2(A_{n-1} + A^{-1}_{n-1})$ converges strongly, $A_0 = I+T$, where $T$ is compact and $||T|| \le 1/2$.

I'm studying for an analysis prelim and am stumped on an old exam problem for which there are no solutions given. The full question is as follows: Let $X$ denote a Hilbert space, and $T$ a ...
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0answers
28 views

Hamiltonian: Compactness

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote the resolvent: $$R(z):=(z-H)^{-1}\in\mathcal{B}(\mathcal{H})$$ Denote compact ...
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0answers
32 views

Are all orthogonal projections conditional expectations?

When will orthogonal projections coincide with conditional expectations? Does that have something to do with the fact that not all closed subspace are probability spaces? Is it why when we fix a ...
2
votes
2answers
26 views

Is a norm closed set(in the topology induced by the norm) weakly closed?

My attempt:Yes because since $T_{\text{Norm}} \supset T_{\text{Weak}} \implies T_{\text{Norm}}^{C} \subset T_{\text{Weak}}^C$ Right? Or have I got something wrong here? This first set inclusion ...
3
votes
5answers
868 views

If a unit ball is compact then why a ball of radius 5 has to be compact too?

So if I use the definition of compactness that every open cover has a finite sub-cover, then as the unit ball is compact , there exists a finite subcover. But if I increase the radius of the ball, why ...
2
votes
1answer
24 views

Example of Non-separable stochastic process.

This question is related to the link: http://www.encyclopediaofmath.org/index.php/Separable_process The link provided a basic definition of separable Stochastic process. I felt all the process under ...
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2answers
24 views

Essential supremum via cumulant

Let $p(t)=\log \mathbb{E}[\exp (tX)]$ for $X$ real valued random variable. Now it holds (assuming that $p$ is smooth and finite on $\mathbb{R}$) that $p'(\infty)=\text{ess}\sup X$. How can I prove ...
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1answer
16 views

Are these two statements involving null sets and $L^2$ Bochner functions equivalent?

Suppose I have two functions $f, g \in L^2(0,T;L^2(\Omega))$ where we have some bounded domain $\Omega$. Suppose that $$\text{for almost all $t$,}\quad f(t) \leq g(t) \quad\text{almost everywhere in ...
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votes
2answers
33 views

Find the rang of $\sin (a) + \sin (b)$ [on hold]

If : $a+b=\frac{\pi }{2}$, Find the range of $$\sin (a) + \sin (b)$$
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0answers
26 views

Semigroup solution via Lumer-Phillips

Let $a$ be a coercive, bounded bilinear form on $H^1(\Omega)$, where $\Omega$ is some sufficiently "nice" region. I defined an operator $A:H^1(\Omega)\mapsto H^1(\Omega)^*$ by: $$ (Au)[v]=a(u,v)\quad ...
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0answers
32 views

Singular Spectrum: Criterion

Problem Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ And its spectral measure: ...
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0answers
12 views

Spectral Measures: Permutability

Given a Hilbert space $\mathcal{H}$. Consider spectral measures: $$E^{(\prime)}:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Denote their operators by: ...
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1answer
75 views

The set of continuity of a pointwise limit of continuous functions

Let $\{x_n(t)\}_{n=1}^{\infty}$ be real a sequence of continuous function from $[0,1]$ to $\mathbb{R}$, and $\{x_n(t)\}_{n=1}^{\infty}$ converges pointwise to $x(t)$ i.e. $\lim_{n \to \infty} x_n(t) ...
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0answers
11 views

Bounded v.s. completely bounded homomorphisms between $L^p$ operator algebras

Take an $L^p$ operator algebra to mean a closed subalgebra $A\subset B(L^p(X,\mu))$ for some ("nice") measure space $(X,\mu)$, $p\in[1,\infty)$. Equip the matrix algebra $M_n(A)$ with the norm ...
1
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1answer
21 views

Operator equation $Au = f$ for $-\Delta u(x)=f(x)$

We consider the boundary value problem on a bounded, open domain $\Omega \subset \mathbb R^n$ determining $u : \Omega \rightarrow \mathbb R$ such that $$-\Delta u(x)=f(x), \qquad ...
1
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0answers
11 views

An example of frame operator.

A sequence of distinct vectors $\{f_1,f_2,...\}$ belonging to a separable Hilbert space $H$ is said to be a Frame if there exist positive contants $A$, $B$ such that, for $A<B$ and for all $f\in ...
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1answer
31 views

What modification is this of the notion of Renyi divergence?

Given two probability distributions $P$ and $Q$ over the same outcome and event space (assume finite if needed) one defines their Renyi divergence as $D_\alpha (P \vert \vert Q) = \frac{1}{\alpha -1} ...
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1answer
19 views

Number of equivalence Relations containing $(1,2)$

Find the number of equivalence Relations on the Set $A=\{1,2,3 \}$ which contains the Element $(1,2)$. My Try: Since $(1,2)$ is to be included, so is $(2,1)$ since the Relation should be Symmetric ...
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1answer
19 views

how to define that a nonlinear operator is bounded and continuous

We always see the definition of bounded and continuous linear operator. I am wondering how to define that a nonlinear operator is bounded and continuous. Is there any book providing this definition?
1
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1answer
41 views

Is $C^{\infty}$ dense in $W^{k,p}$?

The $C_c^{\infty}$ are certainly not sense in an arbitrary $W^{k,p}$ space. Despite, I started wondering whether at least $C^{\infty}$ is dense? Now, this can certainly not be true for general ...
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votes
1answer
28 views

How can I prove that $f$ and $g$ are measurable functions [on hold]

Let we have the following functions : $f(x)=(\sin x)^4$ and $g(x)=(\cos x)^4$ How can I prove that $f$ and $g$ are measurable functions
2
votes
2answers
56 views

Operator on $L^2 (0,1)$ defined by convolution with $|x-y|^{-\alpha}$

Define $A: L^2 (0,1) \to L^2(0,1)$ $$Af(x) = \int_0^1 f(y) \frac{1}{|x-y|^\alpha} dy \quad , \quad \alpha \in (0,1)$$ For what values of $\alpha$ is it well defined? Bounded? Compact? I tried doing ...