2
votes
0answers
20 views

Relationship between eigenvalues of differential operator and eigenvalues of its adjoint operator.

I am considering $L\phi = -\triangle \phi + u \cdot \nabla \phi$ and its "adjoint" operator $L^* \phi = -\triangle \phi - \nabla \cdot (\phi u)$ on a bounded domain $\Omega \subseteq \mathbb{R}^n$. ...
0
votes
0answers
54 views

Real Analysis and dynamics

I am looking for a textbook or similar resource that addresses the content in a rigorous graduate course in real analysis(at the level of Rudin/Royden) with the following criteria: No hand waving - ...
0
votes
0answers
16 views

What is the convergence criterion for linear fixed-point iteration in Banach space?

Consider an iterative process of the form $x^{n+1}=A x^n + b$. When $A$ is a linear operator in $\mathbb R^n$ then the criterion of convergence is $\rho(A)<1$, where $\rho(A)$ is spectral radius of ...
0
votes
0answers
32 views

Riesz Representation Theorem

I am unfamiliar with Quantum Mechanics and all that stuff. I have recently studied Riesz Representation Theorem , I got to know that it justifies ket and the bra notation. Can anyone please give an ...
2
votes
0answers
32 views

uniform equivalence to unit vector basis of $\ell_p$

Let $(e_n)$ be the unit vector basis of $\ell_p$, $1\leq p<\infty$. It is well-known that if $(x_n)\subset\ell_p$ is seminormalized and weakly null then it contains a subsequence equivalent to ...
3
votes
2answers
85 views

Any good, undergraduate level introductions to Functional Analysis?

In my lower division math classes, my instructors referenced functional analysis as essentially the extension of linear algebra to infinite dimensional vector spaces along with some real analysis. As ...
1
vote
0answers
18 views

Conjugating an operator with a gauge transformation; how is the kernel affected.

For the differential operator $$ D := i I \frac{d}{dx} + A(x) \colon C^\infty_T([0,\beta]),\mathbb{C}^m) \to C^\infty_T ([0,\beta],\mathbb{C}^m) $$ where $A(x)$ is Hermitian and $C^\infty_T ...
0
votes
0answers
24 views

General solution of the recurrence equation with real shifts

How to find a general solution of the following functional (recurrence) equation: $$f(x) = c_1 f(x - a_1) + \dots + c_n f(x - a_n), \tag{1}$$ where $c_i, a_i$, $i = 1, ... n$ are arbitrary real ...
3
votes
1answer
34 views

Integration by parts in Bochner Lebesgue spaces.

Does there exist an analogous of integration by parts for expressions such as: $$\int_0^T {\langle u(t),v(t) \rangle }\, \mathrm{d}t,$$ where $u,v\in L^2([0,T];H)$, for some Hilbert space $H$? If so, ...
1
vote
1answer
49 views

Which of these things is not like the others?

What's in a name? Well quite a lot, if you're confused enough. I have an engineering-style mathematics education, based on good old hand waving and learning bits and pieces from all over the place. I ...
2
votes
0answers
30 views

$f_n \rightharpoonup f$ in $L^q(Q)$ $\forall q < \infty$ and $f_n' \rightharpoonup f'$ in $L^2(0,T;H^{-1})$ implies $f_n \to f$

(... in $C^0([0,T]; H^{-1})$. ) Let $f_n$ be a sequence of functions defined on $Q:=(0,T)\times \Omega$, where $\Omega$ is a bounded domain. I have read this: Since $f_n \rightharpoonup f$ in ...
0
votes
1answer
22 views

Reference needed for: $u \in H^1(0,T;L^2)$ if and only if $\int_0^{T-h}\lVert u(t+h)-u(t) \rVert_{L^2}^2 \leq C|h|$

There is a result of the form: a function $u \in H^1(0,T;L^2)$ if and only if $$\int_0^{T-h}\lVert u(t+h)-u(t) \rVert_{L^2}^2 \leq C|h|$$ holds for all $h \in [0,T]$. I have only seen one place ...
0
votes
0answers
57 views

A different weak formulation for parabolic PDE problem (test function space $L^2(0,T;H^2(\Omega))$).

Consider the PDE $$u_t - \Delta u = f$$ $$u(0) = u_0$$. Instead of the usual weak form, let me take this one: for every $\varphi \in L^2(0,T;H^2)$, $$\int_0^T \langle u_t, \varphi \rangle - \int_0^T ...
0
votes
1answer
49 views

Compactness of Sobolev Space in L infinty

I want to show that if $u_{m} \rightharpoonup u$ in $W^{1,\infty}(\Omega)$ then $u_{m} \rightarrow u$ in $L^{\infty}(\Omega)$. I know that I can't directly use the compactness of Rellich Kondrachov ...
0
votes
3answers
93 views

Measure Theory and Functional analysis exercise book

I'm looking for a big collection of exercises of functional analysis and measure theory. I know a lot of theory books which present some excercises (Brezis, Rudin, Lang, Royden, and others) but I was ...
1
vote
0answers
31 views

What does “weakly compact” mean when applied to subsets $X \subset Y$?

Let $X$ be a subset of a Banach space $Y$. Please can you give me a definition of what "$X$ is weakly compact" means? I want one which is in terms of sequences and boundedness, as opposed to one with ...
1
vote
0answers
29 views

A regularity result for a parabolic PDE? Want $u' \in L^\infty((0,T)\times \Omega)$

Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b\quad\text{for all $(x,t)$}$$ $$\frac{dg}{dt} \in L^\infty((0,T)\times ...
0
votes
0answers
18 views

Minimizing the homogenuos Sobolev norm for a given trace

Suppose that $\Omega$ is a bounded domain with regular boundary (think $C^1$). We have a function $f_b:\partial\Omega\to\Bbb R$ and we can expand it to the whole $\Omega$ in the sense of ...
4
votes
0answers
28 views

How do they call the topological tensor product that classifies operators from Hilbert space?

Let $V$ and $W$ be topological vector spaces. There are different ways to complete the tensor product $V \otimes W$, and the only ones that are usually discussed in introductory literature are the ...
0
votes
0answers
32 views

operator onto-theorem

I have this theorem: Let $V$ a Banach space, reflexive,separable, and let $A$ an operator monotonic, bounded, semi-continuos, coercive. Then, $A$ is onto. Where we can find the proof of this ...
1
vote
0answers
28 views

Reference request for nonlinear functional analysis notes.

I'm currently trying to read a paper on Fixed Points of Asymptotic Contractions" by W.A. Kirk. A small excerpt can be seen here. Those with accounts on the Elsevier page can see the whole content. ...
0
votes
0answers
20 views

Summability of Fourier series from Banach space point of view

I am under the impression the following is true (any pointer to a reference would be appreciated ): Theorem (Katznelson?) For any $f \in C[0,1]$ with Fourier coefficients $\{ \hat{f}(n)\}$, there ...
1
vote
1answer
25 views

Patching up basis for $L^2(\mathbb{R}^n)$

Given an orthogonal basis for $L^2(I)$ where $I\subset\mathbb{R}^n$ is the unit cube, can we construct an orthogonal basis for $L^2(\mathbb{R}^n)$ by translations/dilations etc.? Any reference to ...
2
votes
1answer
37 views

Reference on $\mathcal{L}^p(I;X)$

I am doing some reading on evolution equations, and $\mathcal{L}^p$ spaces with functions with values in a Banach space $X$ appears rather often. However I have not found a comprehensive reference ...
3
votes
0answers
41 views

Reproducing kernel Hilbert sapce

I encountered the following claim (verbatim): Theorem Let $V$ be a subspace of $L^2(\mathbb{R})$ and $\{e_n\}$ be a orthonormal basis of $V$. The $V$ is a reproducing kernel Hilbert space with kernel ...
2
votes
1answer
86 views

$H^{-1}(\Omega)$ given an inner product involving inverse Laplacian, explanation required

Let $\Omega$ be a bounded domain and define $V=L^2(\Omega)$ and $H=H^{-1}(\Omega)$. Endow $H$ with the inner product $$(f,g)_{H} = \langle f, (-\Delta)^{-1}g \rangle_{H^{-1}, H^1}$$ where ...
6
votes
2answers
150 views

Popular Topics in mathematical analysis(Functional analysis)

I am writing a text(as a duty by my mentor) dealing with the recently popular topics(including open problems) in mathematical analysis. At first part, I briefly introduced the mathematical ...
3
votes
0answers
40 views

operator on separable banach space whose spectrum and point spectrum is prescribed compact set

I am interested in obtaining the following paper: G. K. Kalisch, "On operators with large point spectrum," Scripta Math. 29 No. 3-4, (1973), 371-378. According to Ben Mathes, "Strictly Cyclic ...
2
votes
2answers
81 views

Normal operators spectral theory

Can anyone guide me to a good resource for proving the spectral theory for normal operators and proving they admit invariant subspaces. When I google it, either it is just the finite dimensional case ...
7
votes
2answers
137 views

Weak periodic solution of parabolic PDE

Take $$ u_t(t) + A(t)u(t) = f(t), $$ $$ u(0) = u(T), $$ where $A$ is an linear elliptic operator and the first equation is an equality in $L^2(0,T;V^*)$ for $V \subset H \subset V^*$ Hilbert triple. ...
5
votes
2answers
97 views

Extending weak solution to global weak solution of parabolic PDE

Fix $T > 0.$ Let $V \subset H \subset V^*$ be a Gelfand triple. Consider the linear parabolic PDE $$u_t - Au = f\quad\text{in $L^2(0,T;V^*)$}$$ $$u(0) = u_0$$ where $u_0 \in H$ and $f \in ...
1
vote
0answers
39 views

Reference for Codimension in Infinite Dimensional Normed vector spaces

There are a couple identities I would like to use related to the codimension and its relationship to the annihilator; some of these seem to be true for all normed vector spaces, and others seem only ...
5
votes
2answers
225 views

Spectral theorem for unbounded self-adjoint operators on REAL Hilbert spaces

In all books that I have checked the spectral theorem (every self-adjoint operator on a Hilbert space is unitary equivalent to a multiplication operator on some $L_2(\mu)$) is only stated for complex ...
0
votes
1answer
27 views

Cross Product Algebras references

Can someone give some references to introductory books or online notes about group algebras and cross-product algebras ? I've already searched on Google (but only for some online notes). The purpose ...
2
votes
0answers
30 views

Various Definitions of Direct Integrals

The Wikipedia article on Direct Integrals gives the following remark on the definition (of the direct integral of Hilbert spaces): This definition is apparently more restrictive than the one given ...
2
votes
1answer
82 views

Reference for Dual Banach Algebras

Can anyone suggest a good reference book to learn about dual Banach algebras? I they come up a lot in things I read but I rarely see more than just an on the fly definition, but not much ...
0
votes
0answers
28 views

Why is the n-fold tensor product of S(R) dense S(Rn) [duplicate]

I need a reference request for the following fact: Every Schwartz function on $\mathbb{R}^n$ is the limit of a series of elements $$ (x_1, \dots, x_n) \rightarrow h_1(x_1) \cdots h_n(x_n),$$ where ...
3
votes
1answer
52 views

A question about extensions of Markov semigroups

I've cross-posted this to MO, if a reply appears on that post I'll update this one. Suppose that $\{T(t)\}_{t\geq 0}$ is a Markov semigroup on the space of continuous bounded functions defined on ...
1
vote
0answers
24 views

Gaussian Smoothing Error and “Hard Analysis” Bounds

Let $p \in (0,\infty)$. Consider a function $f \in L^p([0,1])$, and let $$\phi_\epsilon(x) = \frac{\exp(-x^2/2\epsilon^2)}{\sqrt{2\pi\epsilon^2}}$$ denote a $0$-mean Gaussian of variance $\epsilon$ ...
3
votes
1answer
29 views

nonsurjectivity of Banach-Stone theorem

Currently I'm studying the extension of Banach-Stone theorem using this book. There is one section about the removal of surjectivity of the isometric isomorphism between the two function spaces $C(K)$ ...
2
votes
3answers
174 views

C* Algebra textbook recommendation

I have read the first two chapters from Analysis Now and the chapter on C* algebras (chptr 8?). I'm taking a course on C* algebras in the spring and am currently overwhelmed with the choices. I'd ...
1
vote
0answers
21 views

Some basic questions about $C^k(I \times A)$

Let $I=[0,T]$ and $A$ be a bounded set that may or may not be compact. Let $f=f(x,t) \in C^k(I \times A)$. Am I right: 1) $f_x, f_t \in C^{k-1}(I \times A)$ 2) $f_{xt}, f_{tx}, f_{xx}, f_{tt} \in ...
7
votes
2answers
506 views

$L^1$ and $L^{\infty}$ are not reflexive

I want some proof for the following statement : $L^1$ and $L^{\infty}$ are not reflexive. Can anyone help me, please? or reference me?
4
votes
0answers
78 views

Can the “inducing” vector norm be deduced or “recovered” from an induced norm?

Can the "inducing" vector norm be deduced or "recovered" from an induced (operator) norm? This question occurred to me after seeing this question. I'm hoping that perhaps there exists something like ...
2
votes
0answers
112 views

Functional analysis problem involving maximal ideal space

For the past week, I've been trying to solve, as a practice homework exercise, Problem 6 of Chapter 11 in Rudin's Functional Analysis, but have not gotten very far it seems. The problem is as follows: ...
0
votes
1answer
19 views

$K$ and $L$ homeomorphic, then $C(K)$ is isomorphic to $C(L)$

Can someone sketch the proof (or give me some reference) of the following fact : If $K$ and $L$, compact and Hausdorff spaces, are homeomorphic then the lattices $C(K)$ and $C(L)$ are isomorphic. (I ...
3
votes
1answer
116 views

Textbook for functional analysis in the style of Amann/Escher

most textbooks I've seen so far are not concise enough for my taste and try to give way too much motivation. Or they're written with a too large focus on applications... Rudin wasn't bad contentwise, ...
5
votes
1answer
87 views

What is Newton's theorem?

I'm reading a paper about mathematical physics at the moment and am wondering about the following: Let $w\colon\mathbb{R}^2\to\mathbb{R}$ be defined by $w(x)=-\log|x|$ and ...
2
votes
1answer
38 views

Tempered fundamental solutions

According to the Malgrange–Ehrenpreis theorem every nontrivial linear constant coefficient PDO $P(\partial)$ admits a fundamental solution $E\in\mathscr{D}'$; I wonder whether $P(\partial)$ admits a ...
7
votes
3answers
1k views

Functional analysis textbook (or course) with complete solutions to exercises

I am a Ph.D. student in economics and I plan to study functional analysis by myself either this winter or the next summer. I am currently looking for a textbook, and since I am studying it by myself, ...