All Questions
10,826 questions
10
votes
7
answers
1k
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Constructive proof of existence of non-separable normed space
I am looking for a constructive proof of one of the following two statements. If they are not constructively provable, I would be very thankful for an explanation as to why that is so (i.e., at which ...
- 133
10
votes
1
answer
1k
views
Separating vectors for C$^*$-algebras
(I asked this on math.stackexchange, without response).
Let $A$ be a C$^*$-algebra, concretely acting on a Hilbert space $H$. Suppose that $\xi_0\in H$ is cyclic and separating for $A$ (that is, the ...
- 18.7k
10
votes
5
answers
4k
views
Orthonormal basis for non-separable inner-product space
Suppose X is an inner product space, with Hilbert space completion H (actually, I'm interested in the real scalar case, but I doubt there's any difference). If H is separable, then so is X, and I can ...
- 18.7k
10
votes
5
answers
1k
views
What is a rigorous statement for "linear time-invariant systems can be represented as convolutions"?
In Signal Processing books, a fundamental theorem is that linear time invariant systems can be represented as a convolution with a distribution. Could you give a mathematically rigorous statement of ...
- 2,745
10
votes
1
answer
368
views
Group of isometries of Banach spaces a topological group?
Let $X$ be a Banach space and let $\mathrm{Iso}(X)$ be its group of isometries, i.e., the set of surjective linear maps $T: X \to X$ with $\|Tx\| = \|x\|$.
Q: Is $\mathrm{Iso}(X)$ a topological group ...
- 9,853
10
votes
1
answer
574
views
General validity of separation of variables
Let $L$ be any differential operator (not necessarily linear).
Given initial conditions and boundary conditions (of any type), I am interested in general statements of the form:
Given a boundary ...
- 237
10
votes
1
answer
594
views
Are the polynomials in $\{1/t\}$ dense in $L^2(0,1)$?
Added. My question in the title was solved (in the negative) by Nik Weaver (in the answer below) and Mateusz Kwaśnicki (in the comments). In both solutions, the reason is that the $L^2$ density fails ...
- 13.8k
10
votes
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
809
views
An extremal problem related either to an uncertainty principle on the circle, or else to the prime number theorem
Consider for $X = 1,2, \ldots$ the set $\mathcal{S}_X$ of trigonometric polynomials $f(t) := \sum_{|k| \leq X} c_k e^{2\pi i kt}$ on the circle $\mathbb{T} := \mathbb{R}/\mathbb{Z}$ of degree $\leq X$ ...
- 13.8k
10
votes
1
answer
929
views
Non-probabilistic proof of the Johnson–Lindenstrauss lemma
The Johnson–Lindenstrauss lemma states that a small set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are ...
- 225
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
1
answer
776
views
Saito-Wright definition of Rickart C*-algebras
A C*-algebra is Rickart if for each $x\in A$ there is a projection $p\in A$ so that
$R(x)=pA$.
Here the right-annihilator $R(S)$ of $S\subset A$ is defined
as $$R(S)=\{a\in A\mid xa=0\, \forall x\...
- 810
10
votes
1
answer
521
views
About Friedrichs historical contribution to QFT cited in Reed and Simon
In the Reed and Simon book, Appendix X.7, they mention that Friedrichs provided the first examples of inequivalent representations of the canonical commutation relations via the Weyl relations in the ...
- 424
10
votes
1
answer
693
views
Rigorous proof of the pentagon identity
I briefly recall the statement of the pentagon identity in quantum dilogarithm and cluster algebra.
For $b\in\mathbb{C}$ with $\operatorname{Re}(b)>0,\operatorname{Im}(b)\geq0$, Faddeev, Kashaev ...
- 1,391
10
votes
2
answers
504
views
Generalizations of the Robbins lemma and Gaussian integration by parts
This is getting no attention, so I'll try this here:
The Robbins lemma, named after Herbert Robbins, says that if $X\sim\operatorname{Poisson}(\lambda)$ and $g$ is a function for which $\operatorname{...
10
votes
3
answers
1k
views
Historical developement of analysis and partial differential equations (especially in the 20th century)
Q: Is there a set of some comprehensive surveys or monographs describing (in
technical detail) the historical development of the various
subareas of analysis and partial differential equations?
I'...
10
votes
1
answer
1k
views
Dual space of continuous Banach-space-valued functions
Let $X$ be a Banach space and $K$ some compact Hausdorff space. I am interested in the dual space of the Banach space
$$C(K; X) = \lbrace f: K \to X, \ f \text{ is continuous}\rbrace, \qquad \lVert ...
- 381
10
votes
2
answers
281
views
Weak* continuity of positive parts
I'm a little embarrassed to be asking this, but surely there is a simple argument that I didn't see?
Let $(f_\lambda)$ be a net in $l^\infty$ which converges weak* to $f \in l^\infty$. We do not ...
- 42.8k
10
votes
1
answer
373
views
Real rank zero of group $C^*$-algebras
The concept of real rank zero of a $C^*$-algebra is introduced as non-commutative analogue of dimension ( topological dimension ). For example, it shown (by Brown-Pedersen) such that, if $X$ is a ...
- 399
10
votes
2
answers
1k
views
The intuition behind the Hilbert projective metric and the Perron Frobenius Theorem
Recently I have read a proof of the Perron Frobenius Theorem for positive aperiodic matrices. In this proof, the trick is to put a metric in the "positive quadrant" of $\mathbb{R}^n$, $\mathbb{R}^{n}_+...
user11178
10
votes
2
answers
926
views
Isomorphisms between spaces of test functions and sequence spaces
I am in the process of writing some self-contained notes on probability theory in spaces of distributions, for the purposes of statistical mechanics and quantum field theory. Perhaps the simplest ...
10
votes
1
answer
2k
views
Counting norms on an infinite dimensional vector space
It is known that whenever E is a finite dimensional real vector space, there is only one norm on E up to equivalence (actually one non discrete vector space topology).
Is it known what happens when E ...
- 101
10
votes
1
answer
349
views
On a variant of Carlson’s theorem
My question is on whether or not there exists some monotone strictly decreasing sequence of positive numbers $c_1>c_2>\ldots$ such that given any $f$ which is a uniformly bounded holomorphic ...
- 4,143
10
votes
1
answer
1k
views
The supremum value of $\int f(t) \log{\frac{1}{|t|}} \, dt$ for normalized Fourier pairs non-negative outside of $[-1,1]$
Observe that for any Schwartz function $f \in \mathcal{S}(\mathbb{R})$ having
$$
f(0) = \widehat{f}(0) = 1
$$
and
$$
f, \widehat{f} \geq 0 \quad \textrm{outside of} \quad [-1,1],
$$
the following ...
- 13.8k
10
votes
1
answer
3k
views
Trace of integral trace-class operator
I have seen many answers to the converse question (which seems to be difficult in general), but I would like to ask the following:
Let $T: L^2 \rightarrow L^2$ be a trace-class operator that is also ...
user69109
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
494
views
Optimal exponent in the Lojasiewicz-Simon gradient inequality
Lojasiewicz's theorem asserts that if $F: \mathbb{R}^n\to \mathbb{R}$ is a real-analytic function in a neighborhood of its critical point $0$, then there exist constants $\theta\in (0,1/2]$, $\gamma\...
user81607
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
votes
2
answers
2k
views
When is a space of measures a measurable space?
Let $X$ denote a measurable space, that is, a set equipped with a $\sigma$-algebra $\Sigma(X)$. Let $M(X)$ denote the space of real-valued measures over $X$. This is a vector space over the real ...
- 8,512
10
votes
4
answers
1k
views
References: Infinite dimensional Lie algebras
What I really want are properties (if it is abelian, nilpotent, solvable, simple, or semisimple; Cartan subalgebras...) of the Lie algebra of smooth functions on a symplectic manifold $(M,\omega)$; ...
- 641
10
votes
1
answer
869
views
Complement of a subspace which is a cartesian product
Let $H$ be a Hilbert space and $U$ a closed subspace of $H\times H$ .
Does then exist closed subspaces $V$ and $W$ of $H$ such that $H\times H =
U \oplus (V\times W)$ ?
See also Perturbations of an ...
- 2,753
10
votes
2
answers
1k
views
Are operators with trivial spectrum nilpotent in a sense?
Being far from analysis, I recently learned about the Invariant subspace problem and came up with the following (perhaps simple or well-known) question.
Let $H$ be a separable complex Hilbert space ...
- 32.4k
10
votes
2
answers
843
views
Implicit function theorem with continuous dependence on parameter
Let $X,Y$ be Hilbert spaces and $P$ a topological space$^1$ and $p_0\in P$.
Let $f:X\times P\to Y$ be a continuous map such that
for any parameter $p\in P$, $f_p:= f|_{X\times \{p\}}:X\to Y$ is ...
- 2,533
10
votes
1
answer
439
views
Interpolation between $L_1^0$ and $L_2^0$
Let $L_p^0$ be the mean zero functions in $L_p(G)$, where, say, $G$ is an infinite compact group endowed with normalized Haar measure. Suppose that $T$ is a bounded linear operator on $L_1$ that maps $...
- 31.5k
10
votes
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
755
views
The Dirichlet heat semigroup, $L^1_\delta$, and the dimension shift phenomenon
In relation to the question on the Hardy inequality and the answer by Terry Tao, I've always been curious about the following:
Let $U \subset \mathbb{R}^n$ be a bounded domain of class $C^2$, $(e^{-t ...
10
votes
1
answer
314
views
Weakly metrizable sets in normed spaces
A similar question was asked on MSE without getting an answer.
In the proof of lemma 1.2 of Asplund operators and holomorphic maps the author (my attempt to contact him failed because the only e-mail ...
- 16.4k
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
366
views
Are all compact subsets of Banach spaces small in a measure-theoretic sense?
Definition. A subset $K$ of a topological group $X$ is called measure-continuous if there exists a $\sigma$-additive Borel probability measure $\mu$ on $X$ such that for every compact subset $C\subset ...
- 41.9k
10
votes
1
answer
700
views
Reference request: Riesz potential $I_\alpha : L^{d/\alpha} \to \rm{BMO}$?
Let us denote the Riesz potential in $\mathbb R^d$ by
$$
I_\alpha (f)(x) := c_{d, \alpha} \int_{\mathbb R^d} \frac{f(y)}{|x-y|^{d-\alpha}}
\, dy.$$
By the classical Hardy-Littlewood-Sobolev theorem ...
- 632
10
votes
1
answer
591
views
Grothendieck spaces and total subspaces of the dual
There is probably an embarrassingly simple counter-example to my question but I couldn't figure it out myself. Let me give it a try here.
A Banach space $X$ is Grothendieck if weak*-convergent ...
- 11.3k
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
680
views
A problem concerning $L^2([0,1]\times[0,1])$
Trying to solve a conjecture in differential geometry, I am leaded to the following problem (which may seem weird to a analyst). I wonder if anyone know some techniques that happen to solve it.
Let $...
- 1,804
10
votes
1
answer
454
views
Open Questions about Wasserstein Space and PDE
While working on my thesis, I encountered the idea of OMT and started reading some more (like Villani's book). In particular, I came across a PhD thesis by Martial Agueh. I thought it was interesting ...
- 427
10
votes
1
answer
433
views
Shift invariant subspaces of $l^1$
There is a simple characterization of shift-invariant closed subspaces of $l^2$: for any measurable subset $S$ of $\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}$, the set of elements of $l^2$ whose Fourier ...
- 42.8k
10
votes
2
answers
960
views
Stone-Weierstrass for cones
A version of the Stone-Weierstrass Theorem asserts: If A is a linear subspace of C(K), the set of continuous functions on a compact space, and if A is a subalgebra that contains the constant functions ...
- 101
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
1
answer
515
views
Complemented subspaces in the dual of James' space $J$
James' space $J$ is subprojective; i.e., every infinite dimensional (closed) subspace of $J$ contains an infinite dimensional subspace which is complemented in $J$. This fact can be found in Corollary ...
- 4,461
10
votes
1
answer
573
views
Tannakian formalism for topological Hopf algebras
Tannaka-Krein duality allows, under the appropriate assumptions, to reconstruct a Hopf algebra from its category of modules. This method was found to be powerful for instance in the work of Etingof-...
- 1,609