All Questions
Tagged with reference-request fa.functional-analysis
1,021 questions
12
votes
2
answers
2k
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Reference on Minty's trick
I am searching for a precise reference for the following result:
Consider $f:\mathbb{R}_+\rightarrow\mathbb{R}_+$ a nondecreasing function.
Assume that a sequence of nonnegative functions $(u_n)_n$ ...
- 3,425
12
votes
1
answer
217
views
A variant of $\ell^2$-cochains
Suppose $X$ is an infinite countable CW complex which satisfies the following property: for all $k$-cells $e$, the number of $(k+1)$-cells incident to $e$ is at most $c_k$, where the latter is some ...
- 18.8k
11
votes
2
answers
1k
views
Do non-stable Banach spaces exist?
Let $K$ be $\mathbb{R}$ or $\mathbb{C}$. A Banach space $X$ over $K$ is stable if $X\cong X\times K$. I encountered the following question in some papers in the sixties:
Is every infinite ...
- 7,583
11
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1
answer
441
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Example of Banach spaces with non-unique uniform structures
While it is known that compact Hausdorff spaces admit unique uniform structures, it is further shown by Johson and Lindenstrauss's result that Banach spaces are characterized by their uniform ...
- 8,071
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4
answers
2k
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Spectral theorem for unbounded self-adjoint operators on REAL Hilbert spaces
This question was posed on MathStackExchange but did not get an answer (even with a bounty).
In all books that I have checked the spectral theorem (every self-adjoint unbounded operator on a Hilbert ...
- 16.4k
11
votes
1
answer
654
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Nonseparable Hilbert spaces as quotients of spaces of bounded functions
Is the following result true: the Hilbert space $\ell^{2}\left(2^{\Gamma}\right)$ is a quotient of $\ell^{\infty}\left(\Gamma\right)$ for any
uncountable $\Gamma$ ? [I think it is, but cannot remember ...
- 4,060
11
votes
1
answer
1k
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Quasi-nilpotent trace class operators as limits of nilpotents
In as yet unwritten work with T. Figiel and A. Szankowski we make an observation in a Banach space context that for Hilbert spaces reduces to:
If $T$ is a quasi-nilpotent (i.e., has only $0$ in its ...
- 31.5k
11
votes
1
answer
336
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Notions in the literature capturing the "symmetric" or "homogeneous" flavour of $L_p$?
This post/question is admittedly vague, but I hope that with some feedback in comments it could be made more precise.
For $E$ a Banach space, $K(E)$ and $B(E)$ will denote the Banach algebras of ...
- 25.8k
11
votes
1
answer
486
views
Resources for divergent / asymptotic series
This series is divergent; therefore, we may be able to do something with it. -- Oliver Heaviside
[Edit (1/14/21) from the answer by Count Iblis to a recent MO-Q on math vids: An enthusiastic intro is ...
- 10.5k
11
votes
1
answer
229
views
The set of boundary vectors of compact convex body has empty interior
Let $K$ be a compact convex body in the Euclidean space $\mathbb R^n$ and $\partial K$ be its topological boundary in $\mathbb R^n$.
Definition. A vector $\mathbf v\in\mathbb R^n$ is called $K$-...
- 41.9k
11
votes
1
answer
691
views
Reference request: Fourier transform on the multiplicative group of real numbers
Let us consider the three groups $(\mathbb{R},+)$, $(\mathbb{Z}/2\mathbb{Z},+)$ and $(\mathbb{R}^\times,\cdot)$ (where $\mathbb{R}^\times := \mathbb{R} \setminus \{0\}$). We endow $\mathbb{R}$ with ...
- 12.6k
11
votes
0
answers
344
views
Tauberian Theorem for 1-parameter groups of operators
The Wiener Tauberian Theorem gives condition on an $f\in L^1(\mathbb{R})$ such that the "induced 1-parameter family" $\{T_b(f)\}_{b\in \mathbb{R}}$ has a dense span in $L^1(\mathbb{R})$; ...
- 5,405
10
votes
2
answers
489
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Surjective linear isometries on $\ell_\infty(\mathbb{N})$
In Volume 1 of "Classical Banach Spaces" Lindenstrauss and Tzafriri note that all surjective linear isometries on $\ell_\infty$ are of the from $(a_i) \mapsto (\varepsilon_i a_{\pi(i)})$ ...
- 1,073
10
votes
3
answers
1k
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References: spectral analysis of the Laplacian operator
I'm looking for several references on the spectral analysis of the Laplacian operator. It is such a well-known topic, but I'm a bit struggling to locate modern systematic expositions in the literature....
user89890
10
votes
2
answers
1k
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What function is this? -Counterexample found: it is not Lipschitz-
THE FRAMEWORK
Let $0<\lambda\le1$ and consider
$$
\Psi:(\Bbb R[X]_0,||\cdot||_{\lambda})\longrightarrow(\mathcal C^{\lambda}[0,1],||\cdot||_{\lambda})
$$
defined as
$$
\Psi(p):=\sup_{0\le u\le\...
- 779
10
votes
2
answers
594
views
Existence of a strongly continuous topologically irreducible representation of a compact group on an infinite dimensional Banach space?
Does there exists a triple $(G, X, \pi)$, where $G$ is a compact group, $X$ an infinite dimensional Banach space over $\mathbf{C}$, and $\pi : G \to B(X)$ a strongly continuous representation of $G$, ...
- 960
10
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1
answer
915
views
Density-$c_0$ in $\ell^\infty$
Let $A \subseteq \mathbb{N}$, define the upper density of $A$ as,
$$
\overline{\delta}(A) := \limsup_{N\to\infty}\frac{|A\cap\{1,2,3,\cdots,N\}|}{N}.
$$
This naturally leads to a weaker form of ...
- 203
10
votes
1
answer
652
views
Extending state space to make a process Feller
Let $X$ be a locally compact Hausdorff space, and let $Y_t$ be a continuous Markov process on $X$ with transition function $P(t, x, \Gamma) := \mathbb{P}_x (Y_t \in \Gamma)$. Let $T_t$ be the ...
- 29.8k
10
votes
3
answers
1k
views
Historical developement of analysis and partial differential equations (especially in the 20th century)
Q: Is there a set of some comprehensive surveys or monographs describing (in
technical detail) the historical development of the various
subareas of analysis and partial differential equations?
I'...
10
votes
1
answer
1k
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Dual space of continuous Banach-space-valued functions
Let $X$ be a Banach space and $K$ some compact Hausdorff space. I am interested in the dual space of the Banach space
$$C(K; X) = \lbrace f: K \to X, \ f \text{ is continuous}\rbrace, \qquad \lVert ...
- 381
10
votes
1
answer
802
views
Restriction of irreducible unitary representation to normal subgroup of finite index
Let $G$ be a Lie group (or more generally a locally compact group), let $N$ be a closed and normal subgroup of $G$ of finite index. Let $H$ be an infinite dimensional complex Hilbert space, and let $\...
- 333
10
votes
1
answer
598
views
What happens if we rotate the kernel of an integral operator?
Given an integral operator $K$ on $L^2(\mathbb R)$ with kernel $k(x, y)$, consider the integral operator $L$ on $L^2(\mathbb R)$, whose kernel has the form $k(\alpha x+\beta y, \gamma x+\delta y)$, ...
- 452
10
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1
answer
783
views
When do tensor products of C*-algebras commute with colimits?
Let $I$ be a filtered poset, which you should think of as being huge. Let $A_i$ be an $I$-diagram of $C^{\star}$-algebras and let $A$ be the colimit of this diagram; if necessary, we can also assume ...
- 976
10
votes
1
answer
203
views
Verbal description, or terminology, for the ${\mathcal L}_p$-spaces of Lindenstrauss and Pelczynski
This question is intended for Banach-space specialists and so I will not repeat all the definitions here. My aim is to find out how the Banach space community refers to such spaces in discussions, and ...
- 25.8k
10
votes
1
answer
503
views
Is there an "exponential law" for differentiable maps between smooth manifolds?
Although it seems like a textbook question, I was not able to find a textbook or even a research article answering the following question:
Let $M$, $N$ and $P$ be finite-dimensional smooth manifolds ...
- 843
10
votes
1
answer
593
views
Density of smooth function in Hilbert spaces
I am looking for a simple reference to the following fact:
If $f:\Omega\to\mathbb{R}$ is continuous, where $\Omega\subset H$ is an open subset of a separable Hilbert space $H$, then for any $\...
- 28k
10
votes
1
answer
429
views
Direct sums of operator spaces
I am interested in the $\ell^1$ analogue of direct sums for Operator spaces, e.g. Operator Space Dictionary. Briefly, and operator space is either a concrete subspace of $B(H)$, the operators on a ...
- 18.7k
10
votes
0
answers
422
views
Upper bound Hölder norm of the solution to the non-linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (u (t, x))|^2 u(t, x) \}$
We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb R \to \mathbb R$ belong to the Hölder space $C^{1, \alpha}_b (\mathbb R)$ for some $\alpha \in (0, 1)$. Let $u : \mathbb T \times \...
- 835
10
votes
0
answers
656
views
“Taylor series” is to “Volterra series” as “Laurent series” is to _________?
Preamble
My question is similar to an earlier MathOverflow question:
“Taylor series” is to “Volterra series” as “Padé approximant” is to _________? which I just answered (hopefully my first ever ...
- 446
10
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0
answers
325
views
Are ideals in separable C*-algebras complemented subspaces?
Let $A$ be a separable C*-algebra and $J\subseteq A$ a closed two-sided ideal. Does this make $J$ into a complemented subspace of $A$? In other words, does the quotient map $A\to A/J$ have a ...
- 6,406
10
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0
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761
views
Reference request : Grothendieck's topological space valued integral
As I am learning the different kind of Banach space valued integrals (Pettis, Bochner), I know that Grothendieck made a "mémoire" in his youth about this topic, but I don't know if it is available ...
- 1,616
9
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4
answers
4k
views
Is the space of Radon measures a Polish space or at least separable?
Background: I work on a SPDE problem where in order to apply Prokhorov's theorem I need that some measure space is Polish space. And additionaly it would be good if that space is Banach space. Earlier ...
- 657
9
votes
1
answer
2k
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Is a manifold generically real analytic (with generic real analytic metric)?
I have heard it said in some differential geometry talks that "the generic situation in such and such case is real analytic". My question is, is the generic smooth manifold also real analytic in some ...
- 123
9
votes
4
answers
905
views
Defining the value of a distribution at a point
Let $\omega\in D'(\mathbb R^n)$ be a distribution and $p\in \mathbb R^n$. If there is an open set $U\subset \mathbb R^n$ containing $p$ such that $\omega|_U$ is given by a continuous function $f\in C(...
- 1,942
9
votes
3
answers
3k
views
The classical Krein-Rutman theorem
The classical Krein-Rutman theorem states that any positive compact linear endomorphism $T:X \to X$ on a Banach space $X$ with positive spectral radius $r(T)$ has an eigenvalue $r(T)$ with a positive ...
- 1,202
9
votes
2
answers
611
views
When does $\left\Vert f(\mathbf{N}) - f(\mathbf{M})\right\Vert_{\mathrm{op}} \leq k\left\Vert \mathbf{N} - \mathbf{M}\right\Vert_{\mathrm{op}}$ hold?
Define the Frobenius norm of a matrix as $\left\Vert A \right\Vert_{\mathrm{F}}=\sqrt{\sum_{i,j} A_{ij}^2}$ and the operator norm as $\left\Vert A \right\Vert_{\mathrm{op}}=\sup_{x \not = 0} \frac{\...
- 93
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1
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777
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Abstract result on partitions of unity?
A motivation: The classical Stone-Weierstrass theorem says that polynomials are dense among continuous functions (say, on the unit interval), while the abstract Stone-Weierstrass theorem (and also the ...
- 2,479
9
votes
1
answer
429
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Is $\mathcal{S}(\mathbb{R}^n)$ a tame Fréchet space?
Hamilton's paper "The Inverse Function theorem of Nash and Moser" (1982, Bull. Amer. Math. Soc, vol. 7, n. 1, page $137$) proves that $C^{\infty}(M)$ is a tame Fréchet space when $M$ is a compact ...
- 370
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3
answers
2k
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Generalizations and relative applications of Fekete's subadditive lemma
Fekete's (subadditive) lemma takes its name from a 1923 paper by the Hungarian mathematician Michael Fekete [1]. A historical overview and references to (a couple of) generalizations and applications ...
- 10.5k
9
votes
1
answer
2k
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Density of smooth functions on Hölder spaces
The following result is often cited without reference in the context of PDEs:
Let $\varOmega \subset\mathbb R^n$ be a bounded open set with smooth boundary. If $0<\beta<\alpha<1$ then $C^\...
- 727
9
votes
2
answers
2k
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Nice Classes of Non-Closable Operators
The only thing I know about non-closable operators can be summarised as "they exist, but they're nasty, so let's not talk about them!" This seems to be the case with everyone else I've talked to. I'd ...
- 1,411
9
votes
2
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706
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Measures whose projections are absolutely continuous
Since my question was not answered on MSE, I would like to ask it here.
Let $\mu$ be a finite Borel measure on the plane. Does there exist a characterization of the property that almost all (wrt ...
- 452
9
votes
1
answer
1k
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A question concerning separate and joint continuity of bilinear maps
Suppose that $V$ is a locally convex topological vector space and $f:V^2 \to V$ is a bilinear map. Suppose that $C \subseteq V$ is compact and convex, $f$ maps $C^2$ into $C$ and
$f \restriction C^2$ ...
- 3,547
9
votes
2
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418
views
Reference request: Parabolic Equations
I am a PhD student working mainly on Elliptic Equations. With the other PhDs of my department, we organised a reading group, meaning that we agreed on a book we were all interested in, we meet weekly ...
- 452
9
votes
2
answers
516
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Why operator systems?
A $\mathrm{C}^*$-algebra $\mathcal{A}\subset B(\mathsf{H})$ is a norm-closed, self-adjoint subalgebra of bounded operators on a Hilbert space. If we then take a unital self-adjoint (possibly closed) ...
- 1,037
9
votes
1
answer
202
views
Literature request: Schatten class difference of semigroups
Let $\mathcal{H}$ be a Hilbert space and $A,B$ two operators on it (not necessarily self-adjoint) such that $A, A+B$ are generators of strongly continuous one parameter semigroups $e^{-tA},e^{-t(A+B)}$...
9
votes
2
answers
907
views
When is a mapping the proximity operator of some convex function?
Is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which are proximity operators (in the sense of Moreau) of l.s.c (extended) real-valued functions ?
That is, given $p : ...
- 6,853
9
votes
1
answer
2k
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When will the supporting hyperplane of a convex set coincide with the tangent?
Due to the supporting hyperplane theorem, a convex set $C$ in a separable topological space has supporting hyperplance at each of its boundary points. The theorem only guarantees its existence, now I ...
- 8,071
9
votes
1
answer
1k
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Traces of Sobolev spaces
Is there a simple proof of the following fact?
Theorem. Let $\Omega\subset\mathbb{R}^n$ be a bounded and smooth domain. If $n>2$, then $W^{1,n-1}(\partial\Omega)\subset
W^{1-\frac{1}{n},n}(\...
- 28k
9
votes
3
answers
2k
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Real analyticity of solution of heat equation
Consider the heat equation $\partial_t u - \Delta u = 0, u(0, x) = u_0$ on a complete (non-compact) Riemannian manifold $M$, may be even $\mathbb{R}^n$. I was wondering, what are some known sufficient ...
- 1,407