All Questions
9,958 questions
8
votes
2
answers
865
views
frechet manifolds book
hi, does anyone know a good book or some lecture notes on the theory of frechet manifolds ?
- 30.3k
2
votes
1
answer
608
views
Stein's extension operator and wave front sets
Let $K\subset\mathbb{R}^d$ be a compact set with non-empty interior and Lipschitz boundary. In Section VI.3 of his book "Singular Integrals and Differentiability Properties of Functions", E. M. Stein ...
- 585
7
votes
2
answers
1k
views
A book on Banach Manifold for a Dynamicist
Hi all,
Could you give me a suggestion of suitable book about Banach Manifolds for someone that have background in functional analysis at the level of Conway's book and Do Carmo's book on Riemannian ...
- 4,306
5
votes
2
answers
3k
views
Diagonalization of a matrix of differential operators
Dear community,
i have a question regarding differential operators acting on vector valued functions and how to "diagonalize" them.
To explain my question i will use an example:
Let $V^k$ be the ...
- 47.3k
2
votes
1
answer
949
views
Hereditarily indecomposable Banach spaces and Separable Quotient problem
A Banach space $X$ is called indecomposable if there exists no infinite-dimensional subspaces $M$ and $N$ such that $X = M \bigoplus N$. If every infinite-dimensional closed subspace
of $X$ is ...
CommunityBot
- 1
4
votes
1
answer
1k
views
Hausdorff dimension of graphs .
Is there an easy way to calculate the Hausdorff dimension of the graph of a real "elementary" function, like $f(x)=\sin(1/x)$ ?
- 7,857
6
votes
0
answers
3k
views
Projective and injective tensor product
It is well known that for arbitrary Banach spaces $X$ and $Y$ we have that the dual space
$(X \hat{\otimes}_{\pi} Y)^* = \mathcal{L}(X, Y^*)$.
If we take $\ell^p$ and $\ell^q$ such that $p < q^{\...
- 145
4
votes
1
answer
690
views
What does $L^\infty_\varepsilon$ mean?
In Volume 4 of Reed and Simon on page 83 the authors refer to the space $(L^\infty(\mathbb{R}^3))_\varepsilon$,
and later on page 119 they use $L^\\infty_\varepsilon$.
Are these two spaces the same? ...
CommunityBot
- 1
4
votes
1
answer
471
views
Embeddings for spaces of maximal regularity
Let $T\in(0,\infty)$ and $\Omega\subset\mathbb R^n$ be a smooth domain. In terms of maximal regularity it can be very beneficial to know for which $s_i,p,n$ the following holds true
$W^{s_1,p}(0,T;L^...
- 316
3
votes
4
answers
514
views
Better terminology than "equivalence class of functions"
Let $X = C(\mathbb R)$ be the Fréchet space of real-valued continuous functions. For each $f \in X$ and each compact set $D \subseteq \mathbb R$, let $$[f]_D = \{ g \in X : \mbox{$g(t) = f(t)$ for ...
- 17k
6
votes
2
answers
979
views
Literature on behaviour of eigenfunctions under multiplication?
Dear community,
I would be happy about any literature or comments on the behaviour of the pointwise product of eigenfunctions of a self-adjoint operator with discrete spectrum, acting on a separable ...
- 45.7k
3
votes
1
answer
624
views
How to calculate a Fredholm index numerically
How can one calculate the index of a Fredholm operator numerically ?
In numerically calculations one uses always finte dimensional spaces.
But linear operators on finite dimensional spaces have ...
- 3,343
1
vote
1
answer
247
views
Distance between lattices of invariant subspaces of matrices
For a linear transformation $A: C^n \to C^n$ let $Inv(A)$ be the lattice of all $A$-invariant subspaces. In work I.~Gohberg, L.~Rodman "On the Distance between Lattices of Invariant Subspaces of ...
- 17k
2
votes
1
answer
323
views
Recovering Schauder decompositions
The problem of Schauder decomposition of a given Banach space seems to play an important role in the geometry of Banach spaces, especially when one is interested in finite dimensional Schauder ...
- 31.5k
0
votes
1
answer
454
views
Is this set of functions compact?
Let $\mathcal{F}$ be the set of continuous functions $\varphi$ from $\mathbb{C}$ to $[0,1]$ that satisfy $\begin{align}\varphi(z)=\frac{1}{2\pi}\int_{0}^{2\pi}\varphi(z+e^{i\theta})d\theta\end{align}$ ...
CommunityBot
- 1
4
votes
2
answers
484
views
When is a metric space isometrically embeddable into some Banach space?
EDIT
Oops---I found the answer to the first question of mine here on Wikipedia---this is really classic material. I'll leave the question open for a bit, in case someone tells me something ...
- 28.6k
9
votes
1
answer
456
views
Embeddings of Sobolev-Orlicz spaces
The Birnbaum--Orlicz spaces generalize the Lebesgue spaces (see http://en.wikipedia.org/wiki/Birnbaum-Orlicz_space for a precise definition). The space $L_\Phi(\Omega)$ is defined for convex functions ...
- 27.5k
4
votes
1
answer
568
views
Crossed product of a non unital C*-algebra
Let $X$ be a locally compact space, and let $T:X\rightarrow X$ be a homeomorphism. Then \begin{align*}
&\alpha:C_0(X)\rightarrow C_0(X)\\\
&\alpha(f)=f\circ T
\end{align*}
is an automorphism. ...
- 43
1
vote
1
answer
312
views
Invertibility of frame/sampling operator on Bargmann-Fock spaces
Let $F_\alpha ^p (\mathbb{C}^n)$ for $1 < p < \infty$ and $\alpha > 0$ be the Bargmann-Fock space defined as the Banach space of entire functions $f$ such that $f(\cdot) e^{- \frac{\alpha}{2} ...
1
vote
1
answer
961
views
Strong topology
Let $E$ and $F$ be a locally convex topological vector spaces (LCS) and let $E^{\star}$ and $F^{\star}$ denote the strong duals of $E$ and $F$, respectively.
A dual of $E^{\star}$ given by the $\beta(...
- 11.3k
3
votes
2
answers
1k
views
Density character and cardinality
Assume that $X,Y$ are infinite dimensional Banach spaces. Is it true that if density character of $X$ is less then or equal to density character of $Y$ then $card X \leq card (Y)$ ?
- 31.5k
5
votes
1
answer
673
views
Unbounded representations of Banach algebras
Can a representation of a Banach algebra be unbounded?
To clarify, I'm not asking about a representation as unbounded operators, but
rather a homomorphism $\pi: A \to B(H)$ for some Hilbert space $H$,...
- 7,329
16
votes
2
answers
4k
views
Usefulness of Frechet versus Gateaux differentiability or something in between.
If you have a function $V: L \rightarrow \mathbb{R}$, where $L$ is an infinite dimensional topological vector space, there are multiple notions of differentiability. For $x,u \in L$, $V$ is Gateaux ...
- 399
4
votes
1
answer
964
views
Convergence of Fredholm determinants
Let $(X_N)_N$ be a sequence of trace class operators acting on, say, $L^2(\mathbb{R})$. What are the minimal assumptions in order to have the convergence of their Fredholm determinant
$$
\lim_N\det(...
- 2,135
8
votes
1
answer
1k
views
Spectra of a Symmetric Toeplitz Operator
For a physics application, I would like to be able to compute the eigenvalues of the linear operator (acting on the Hilbert space $\ell^2$) given by an infinite matrix of the form
$\begin{bmatrix}
...
- 21
3
votes
2
answers
949
views
Reference for proof that $C_b^* = rba$
The following theorem seems to have folk status:
The topological dual of the space $C_b(X)$ of bounded continuous functions on a topological space $X$ is isomorphic to the space $rba(X)$ of finite, ...
CommunityBot
- 1
0
votes
0
answers
155
views
General form of a symplectic map
A symplectic automorphism of a Hilbert space has the form $T=U(\cosh S+J\sinh S)$ for a unitary $U$, an antilinear involution $J$ and a positive operator $S$. In fact a version of this goes through in ...
- 1,411
3
votes
1
answer
556
views
Convergence of mountain pass solutions of $-\Delta u+u=u|u|^{p-2}$
Consider the following equation in $\mathbb{R}^N, N \ge 3$:
$$
(E) \quad -\Delta u +u=|u|^{p-2}u,
$$
where $2 < p < 2^{*} =2N/(N-2)$.
Denote by $J: H^1(\mathbb{R}^N) \to \mathbb{R}$ the ...
CommunityBot
- 1
1
vote
2
answers
687
views
High dimensional beta integral (a typo in Stein's book "singular integrals")
Hello,
When I read Stein's book of Singular Integrals, at p. 118, there is an obvious mistake:
$$
\int_{R^n} |x-y|^{-n+\alpha} |y|^{-n+\beta}=\frac{\gamma(\alpha)\gamma(\beta)}{\gamma(\alpha+\beta)},...
CommunityBot
- 1
0
votes
0
answers
298
views
High dimensional beta integral (question following the previous post)
Hello,
This post is a question following the previous post. In one dimensional case, we have
$$
\int_0^x |y|^{1-\alpha} |x-y|^{1-\beta} d y = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} |...
CommunityBot
- 1
4
votes
3
answers
3k
views
Examples of Banach spaces and their duals
There are many representation theorems which state that the dual space of a Banach space $X$ has a particularly concrete form. For example, if $X = C([0,1],\mathbb R)$ is the space of real-valued ...
- 159
7
votes
1
answer
773
views
Equivalent metrics on Fréchet spaces and Lipschitz maps
Lipschitz maps are defined over metric space as maps $f:(X,d_X) \to (Y,d_Y)$ such that
$$ d\left( f(x),f(x^\prime) \right)_Y \le k d(x,x^\prime)_X \ \forall x,x^\prime \in X, $$
where $k$ is a ...
- 31.5k
7
votes
3
answers
814
views
Preduals of B(E)
For a Hilbert space $H$ it is well known that the algebra $B(H)$ has a unique predual; the Banach space of trace class operators.
If $E$ is a Banach space then is it known whether
$B(E)$ is always a ...
- 18.7k
0
votes
0
answers
388
views
Global index of convexity/concavity of a function
We are looking for a global index of the convexity/concavity of a function.
For concreteness, how can I formalize the intuitive notion that a function $f$ is more convex than $g$ where $f,g:[0,1]\...
- 111
1
vote
0
answers
693
views
A question about an equivalent definition of the Schwartz distribution
Hello,
Does anyone know a reference or proof of the "if" part of the following statement?
$$
\mu\in \mathcal{S}'(R)\quad\text{if and only if}\quad \mu*\alpha\in\mathcal{S}(R),\forall \alpha\in C_c^\...
1
vote
0
answers
378
views
Adjoint operators in LCS
Before my main question let me start with the following notions.
Let $X$ and $Y$ be locally convex spaces and let $T \colon X \rightarrow Y$ be a linear mapping. The adjoint of $T$ is an operator
$T^...
- 145
3
votes
1
answer
1k
views
Self-adjoint bounded operator, resolution of the identity, def. of the diagonal
Let $A$ be a self adjoint bounded linear operator with a continuous spectrum
$\sigma(A)=[a,b]$ which acts on a separable Hilbert space. Let
$E_\lambda$ be its resolution of the identity.
For ...
1
vote
1
answer
562
views
Metrizable dual space
I've got the following questions concerning the theory of locally convex spaces :
Let $X$ be a locally convex metrizable space, what is the necessary and sufficient condition to have its dual $X^*$ ...
- 53.3k
0
votes
1
answer
340
views
Reference for spectral theory of group of linear operators
It is not hard to find the spectral theory of a single unitary operator $U$. This is the spectral theory for a $\mathbb{Z}$-action because we consider $U^n$ for $n\in\mathbb{Z}$. This is clear with ...
- 163
3
votes
2
answers
2k
views
trace norm inequality for positive matrices
If $A, B$ are positive $n \times n$ complex matrices, $n$ some integer, then obviously \begin{equation*} \|ABA\|_\text{tr} = tr(ABA) = tr(A^2 B). \end{equation*}
But can we say there is a constant $...
- 9,486
6
votes
3
answers
2k
views
Sequential topological vector spaces
Since I'm dealing with the distinction between sequential continuous and continuous maps at the moment I came to ask myself once again what can be said about spaces where these two notions agree (...
- 981
4
votes
0
answers
189
views
Boundedness criterion for operators on mixed Lebesgue spaces
Define the mixed Lebesgue space $l_{p,q}$ as the space of all doubly indexed sequences
${\bf a}= (a(i,j))_{i,j\in\mathbb{Z}}$ such that
...
- 979
3
votes
1
answer
785
views
Maximal ideals of some algebras
This question comes up after I read the chapter 7 : Banach algebras and spectrum theory from Conway's book. As we have known that if $X$ is compact space, then all the maximal ideals of $C(X)=\{ f : X\...
- 8,559
7
votes
0
answers
161
views
Seeking reference - criterion for the existence of a positive linear functional on an ordered vector space below a given function
The following surely appears somewhere, I would greatly appreciate a reference. (The aim is to get a measure via Riesz representation, but that has nothing to do with the statement.)
Let $X$ be an ...
- 18.7k
-3
votes
2
answers
768
views
Question on Linear Operators
Let $V$ be a normed infinite dimensional vector space. Let $L: V \longrightarrow V$ be a bounded linear operator. Moreover assume that $L$ is 'locally nilpotent' that is:
$$ \forall v \in V \quad \...
- 118k
8
votes
0
answers
349
views
Finding a dimension-free bound for a certain multiplier on Euclidean space
The following question is indirectly motivated by strong type maximal function estimates. Let $f\in L_{p}(\mathbb{R}^{n})$. For $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ define $m(\xi)$ so ...
- 141
5
votes
3
answers
931
views
References for "different" proofs of the spectral theorem for compact operators
It is with some sort of reverential fear that I've come here to write. I've been reading you for a long time, but writing is another story... In any case, I suppose it is too late now to back out!
...
- 10.5k
13
votes
2
answers
2k
views
What is the relationship amongst all the different kinds of spectra?
The word "spectrum" gets tossed around a lot in mathematics, and there seem to be a number of different concepts to which it applies. There is of course a physical connotation to the word which is ...
- 38.6k
1
vote
1
answer
977
views
Fourier transform of distributions with non-standard test functions
This might be a quite simple question for function analysis standards, but it has some obstacles. I'll try to improve the readability a bit by not using the full tex code. A short motivation:
Given a ...
CommunityBot
- 1
11
votes
4
answers
1k
views
Norm continuous infinite dimenisonal representation of a Lie group
Given a Lie group G and an infinite dimensional Hilbert space $\mathcal{H}$. In the literature I have only encountered the two following notions of a representation $\pi$ of G on $\mathcal{H}$ :
1) $\...
- 23k