Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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7
votes
7answers
634 views

What is the best reference for Spectral theory?

I'm studying Bernard Aupetit: A Primer on Spectral Theory but the textbook we are using is a little bit heavy going for me. Is there a best book to learn about these things? Thank you.
0
votes
0answers
51 views

A question related to the Nikolskii fractional spaces

Consider a Nikolskii space, that is $$ N^{s,p}=\{f\in L^{p}(I, d\ell): \|f\|_{\overline{N}^{s,p}}=\underset{h>0}{\sup}h^{-s}\|\tau_{h}f-f\|_{L^{p}(I_{h})}<\infty \}, $$ where ...
6
votes
1answer
232 views

Is an infinite-dimensional “Lebesgue measure” uniquely determined by a set of positive finite measure?

Let $\mu$ be a probability measure on a subset $C \subset \mathbb{R}^\infty$ of the space of sequences, and assume, for simplicity, that $C$ is closed and convex. We say that $\mu$ admits shifts if ...
2
votes
0answers
96 views

continuity with respect to weak-${\ast}$ topology

Let $V:=V([0,1],R)$ be the space of all cadlag functions defined on $[0,1]$ of bounded variation. Thus any element $v\in V$ determines a signed measure $\nu$ on $[0, 1]$ given by the formula $\nu([0, ...
-4
votes
2answers
107 views

Does the Laplacian commutes with the indicator function [closed]

We define the laplacian operator $\Delta$ with the Neumann boundary conditions on the space $H^2(\Omega)$, where $\Omega$ is an open set of $\mathbb{R}^n$ with a smooth boundary $\partial\Omega$, and ...
3
votes
1answer
112 views

A question about Skorokhod metric

I have a question related to the Skorokhod distance. Let $\Omega:=D([0,1],R)$ be the space of cadlag functions $x$ defined on $[0,1]$. Let $\Lambda$ be the collection of non-decreasing continuous ...
1
vote
1answer
179 views

Smooth curves in a Frechet space

Is the space $C^{\infty}([0,1];C^{\infty}(S^1))$ equal with the space $C^{\infty}([0,1]\times S^1)$ ? I am interested in characterizing the smooth curves in the space $C^{\infty}(S^1)$ where $S^1$ is ...
3
votes
1answer
80 views

Skorokhod distance between $\omega, \omega\circ f_{\varepsilon}$ and $\omega, \omega\circ b_{\varepsilon}$

Let $\Omega:=D([0,1],R)$ be the space of cadlag functions $x$ defined on $[0,1]$. Let $\rho$ be the Skorokhod metric on $\Omega$, see e.g. http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g Now define ...
3
votes
1answer
151 views

A nilpotency question on $C^{*}$ algebras

What is an example of a $C^{*}$ algebra $A$ with the property that: for every nilpotent(Quasi nilpotent) $a$ and for every $n\in \mathbb{N}$, there is a $b$ with $b^{n}=a$. To what extent ...
6
votes
0answers
176 views

What function space does holomorphic functional calculus give us?

Let $A$ be a unital Banach algebra, $U$ be an open subset of $\mathbb{C}$, and $A_U:=\{x\in A:\sigma(x)\subset U\}$. Holomorphic functional calculus says that any holomorphic function ...
4
votes
1answer
82 views

Approximation of sets by sets with regular border

What kind of conditions on a (bounded) set $E \subset \mathbb{R}^{n}$ ensure that it can be approximated from outside/inside by sets with regular border (say Lipshitz or $C^{k}$ conditions) in the ...
2
votes
0answers
99 views

Counterexample for closed graph theorem in unmetrizable case

Consider locally convex vector spaces $X,Y$ and let $A: X\to Y$ be a linear operator with closed graph. Closed Graph Theorem states that if $Y$ is Fréchet then $A$ is continuous. What is a ...
1
vote
0answers
43 views

Representing a Pullback as an Infinite Matrix

Let $M$ and $N$ be manifolds and let $T: M \to N$ be a bijective map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ \mathcal{F}(N,\mathbb{R})$) be the space of all functions from $M$ (resp. $N$) to ...
1
vote
0answers
99 views

Properties of a function from its pullback

Edit: I have now removed the duplication previously referred to. Thank you. Let $M$ and $N$ be smooth manifolds and $T: M \to N$ be a smooth map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ ...
2
votes
1answer
136 views

Inverse Problem for Pullback

Let $M$ and $N$ be smooth manifolds and $T: M \to N$ be a smooth map. Let $ \mathcal{F}(M,\mathbb{R})$ (resp.$ \mathcal{F}(N,\mathbb{R})$) denote the space of smooth functions from $M$ (resp. $N$) ...
0
votes
0answers
57 views

existence of locally translation-invariant Borel measure on Frechet manifolds

It is well known that the only locally finite, translation-invariant Borel measure on an infinite-dimensional, separable Frechet space is the trivial measure. I am wondering about an analogous ...
0
votes
1answer
120 views

Exploiting the Linearity of the Pullback [closed]

Edit: This question has been significantly revised. Some recent developments in computational geometry (for example see http://geometry.stanford.edu//papers/fmfrmbs-obsbg-12/fmfrmbs-obsbg-12.pdf) ...
6
votes
3answers
349 views

Pathological product space norm

Let $X$ and $Y$ be two normed vector spaces and $n(\cdot, \cdot)$ be any norm on $\mathbb{R}^2$. Is it always possible to define a norm on the product vector space $X \times Y$ as $||(x, y)||_{X ...
1
vote
1answer
130 views

Wendel Theorem for center of group algebra

Let $G$ be a locally compact SIN-group. Then $ZL^{1}(G)$ has a bounded approximate identity. I want to prove that the multiplier algebra of $ZL^{1}(G)$ is equal to $ZM(G)$ (center of measure algebra). ...
3
votes
1answer
224 views

A.e. pointwise convergence of L2 functions - counterexample for generalization of Carleson's thm

Let $f_n \in L^2[0,1]$ be an orthonormal sequence and let $c_n \in \mathbb C$ be such that $\sum_{n = 1}^{\infty} |c_n|^2 < \infty$. Does this imply that the sequence $\sum_{n = 1}^{\infty}c_nf_n$ ...
4
votes
1answer
86 views

Reference request: Wasserstein metric spaces for non linear weights/mobility?

There is a very nice theory of gradient flows in metric spaces by Ambrosio, Gigli and Savaré. One particularly important application is the quadratic Wasserstein setting, where the metric space in ...
0
votes
0answers
79 views

SVD of Frechet derivative

This is mainly a reference request. Is there a particular characterization of operators A from a Hilbert space H to itself such that the Frechet derivative A'(u) exists for each $u \in H$ and for any ...
9
votes
1answer
614 views

Counterintuitive consequences of the Hahn-Banach theorem

The axiom of choice has many counterintuitive consequences like the Banach-Tarski paradox. The Hahn-Banach theorem is a consequence of the axiom of choice, but it is weaker. I would like to know ...
7
votes
3answers
369 views

Is there a continuous analogue of Ramanujan graphs?

I think it might help to think of the following definition of a Ramanujan graph - a graph whose non-trivial eigenvalues are such that their magnitude is bounded above by the spectral radius of its ...
2
votes
1answer
306 views

Finding a norm on $ \mathbb{R}^X $ such that the “natural” embedding of a metric space $ X $ in $ \mathbb{R}^X $ becomes an isometry

Let $ (X,d) $ be a metric space and consider the function $ T:X \to \mathbb{R}^X$ such that $ T(x)(y) = 1$ if $ y = x $ and $ 0 $ for all other $ y $. Is there a norm on $ \mathbb{R}^X$ such that $ T ...
0
votes
1answer
74 views

Rank of a sequence of covariance matrices

Let $X_i$ ($i=1, \dots$) be an orthonormal basis for $L^2(\Omega, \mathbb P)$. In particular, it holds that $$\mathbb E[X_iX_j] = \delta_{ij}.$$ Now take $Z\in L^2(\Omega, \mathbb P)$ and define ...
1
vote
1answer
167 views

James $\ell_1$-theorem

This question was asked at MSe before but with no answer. I am struggling with the very last estimate in the proof of James' $\ell_1$-theorem. (Please see below an excerpt from Albiac and Kalton's ...
1
vote
0answers
34 views

Request for reference about bound on zeroes of the Laguerre polynomials

Consider the sequence of polynomials given as, $p^{a}_k (x) = (1 - a \frac{d}{dx})^k x^n $ for some parameter $a>0$ and $k$ being a positive integer. For any positive integer $d$ it seems to be ...
0
votes
0answers
104 views

Existence of a mapping in a nonseparable Banach space

Do there exist a nonseparable Banach space $X$, a mapping $F: X\to X$, and an open nonempty subset $D\subset X$ such that $$ \forall\,E>0 \quad \exists\,\delta>0: \quad \forall\,x,y\in D \quad ...
3
votes
1answer
137 views

Equivalence discrete H^2 Sobolev norms

My aim is showing the equivalence of two discrete Sobolev norms. On $\mathbb{Z}^d$, $d\ge 2$, one defines the discrete derivative in the direction of the coordinate vector $\vec e_j$ as $$ ...
1
vote
0answers
86 views

Difference Quotients Evans

There is a theorem in Evans partial differential equation book as follows: if $u \in W^{1,p}(U)$ then for each compact $V$ in $U$ we have that: $ |D^hu|_{L^p(V)} \leq C |Du|_{L^p(U)} $ for all $ ...
3
votes
0answers
153 views

An upper bound for a average of a function in $L_{p}([0,1))$

Suppose that $ f $ is $ 1 $-periodic and that $ f \in {L^{p}}([0,1) $, where $ p > 1 $. Let $$ (D_{n})_{n \in \mathbb{N}_{0}} = \left( \left\{ I^{n}_{j},~ 1\leq j \leq 2^{n} \} \right\} \right)_{n ...
4
votes
0answers
99 views

The representation-theoretic nature of an operator resolvent

Consider parameter $s$ in definition of $R(s,A)=(s I - A)^{-1}$ where $A$ is a linear operator in a vector space $X$. When $X$ is over $\mathbb{C}$, then $s$ is thought to be a complex number. Now ...
4
votes
0answers
217 views

“Partition” of a smooth function in $\mathbb R^2$

This is a question asking for reference. I have a proof of the following. Let $f=f(x,y)$ be a smooth function in $\mathbb R^2$ which vanishes at the origin. Then there exist smooth functions ...
4
votes
1answer
144 views

Are convolution algebras ever “topologically noetherian”?

For finite groups $G$, we have the group ring $k[G]$, and we can think of $G$-representations as $k[G]$-modules. It is known that for $G$ virtually polycyclic, $k[G]$ is a Noetherian ring, which means ...
1
vote
1answer
133 views

Nonlocal Stefan problems

Has there been much work in the setting of Stefan (or general free boundary) problems with some type of nonlocality? A search on Google and MathSciNet give me only a handful of results which greatly ...
6
votes
1answer
166 views

Triangle inequality for $L^1$-norm with respect to a state

It is well-known that the naive construction of non-commutative $L^p$-spaces is performed only in tracial case. I would like to know if it is really a necessity. To wit, let $\varphi$ be a normal ...
5
votes
0answers
123 views

Examples of topologies compatible with a given dual pair

Let $\langle X, Y \rangle$ be a pair of vector spaces put in duality by a non-degenerate bilinear form $\langle \cdot, \cdot \rangle: X \times Y \to \mathbb{R}$. A topology $\tau$ on $X$ is called ...
-1
votes
1answer
89 views

What is the definition of maximal ε-separated set

Nowadays, I am just studying the book wrote by Joram Lindenstrauss and Yoav Benyamini,i.e. Geometric Nonlinear Functional Analysis. The putfroward "maximal ε-separated set".I really can not understand ...
3
votes
0answers
97 views

Is exponential function in a C*-algebra injective on self-adjoint elements?

I asked this question in stackexchange, but it flashed and disappeared: Let $A$ be a C*-algebra and $\exp(x)=\sum_{n=0}\frac{x^n}{n!}$, the usual exponential function from $A$ into $A$. Is it true ...
0
votes
1answer
115 views

Complementation in tensor products

This question, however looks innocent, looks non-trivial to me. Suppose that $X$ and $Y$ are Banach spaces and let $\alpha$ be any reasonable cross-norm on $X\otimes Y$. Reasonable means that ...
1
vote
0answers
129 views

Applications of composition operators on Sobolev spaces

I wold like to know some examples where composition operators on Sobolev spaces are useful. I'm in the following situation. $L^1_p(D)$ - homogeneous Sobolev space, in other words space of locally ...
3
votes
1answer
149 views

What are the applications of Grillakis Shatah and Strauss paper?

I am studying the following paper. Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), no. 1, 160–197. ...
4
votes
0answers
103 views

Pseudodifferential operators on compact manifolds with boundary

I have heard that the square root of the Dirichlet (or the Neumann) Laplacian is not a pseudodifferential operator on compact manifolds with boundary. The context in which this was said was that ...
3
votes
1answer
169 views

Is this inequality true?

Let $\Omega$ be a bounded domain with smooth boundary. Let $$ S=\{u\in C^2(\overline \Omega): \frac{\partial u}{\partial n}=0 \text{ on } \partial\Omega \}.$$ Fix $\Phi\in S$ with $\Phi(x)>0$ for ...
4
votes
1answer
128 views

Some questions on analytic vectors and the integrability of Lie-algebra representations

I would like to ask a number of questions about the theory of analytic vectors and the integrability of Lie-algebra representations, but before I do so, let me fix the terminology to be used in this ...
4
votes
2answers
263 views

Topological characterization of injective metric spaces

Let $\ (X\ d)\ \,(Y\ \delta)\ $ be arbitrary metric spaces. A function $\ f:X\rightarrow Y\ $ is called a metric map (with respect to the given metrics $\ d\ \delta$) $\ \Leftarrow:\Rightarrow\ ...
2
votes
0answers
109 views

Reference request: functional analysis results used in Taubes paper (1980)

I'm studying Taubes paper 'Arbitrary N-vortex solutions to the first order Ginzburg-Landau equations'. I'm looking for a reference of three following theorems: Let $f(x)$ be a convex funtional ...
1
vote
0answers
45 views

Rate of convergence in narrow convergence

Does anyone help me in the following question? I have a sequence of probability measures $\mu_n$ and know that $\mu_n$ converges narrowly to a probability measure $\mu$. Is there any way to estimate ...
3
votes
3answers
488 views

What should be considered a finite size of an infinite dimensional space? [closed]

I've got a map between two infinite dimensional spaces, $f: A\to B$, where $A$ seems "larger" than $B$. For the sake of conversation let's assume that $A$ is the set of smooth maps $\mathbb R^3\to ...