All Questions
12,780 questions
0
votes
2
answers
205
views
ANR Subsets of banach spaces
I need a reference for conditions on a closed subspace of a Banach space to have the homotopy type of an ANR.
5
votes
0
answers
157
views
Containment of an element to an operator system
This question will probably appeal to people in operator systems theory as it is very much related. However, I'm interested in down-to-earth concrete systems with finite dimensional Hilbert space ...
- 315
8
votes
1
answer
1k
views
Beautiful examples of arc-like continua
A continuum is a nonempty compact, connected metric space.
A continuum $X$ is called arc-like if, for every $\varepsilon>0$, there is a continuous and surjective function $f:X\to [0,1]$ such that $...
1
vote
2
answers
941
views
Alternate definitions of $C^{1,\alpha}$ and $C^{1,\alpha}(\bar{D})$ maps
My question is about the precise definition regarding the following:
Let $f$ be an orientation-preserving $C^1$ diffeomorphism of the unit circle $S^1$. So $f'(b)$ exists and can be thought as a ...
- 1,471
1
vote
1
answer
215
views
About principal values and Wirtinger derivative
Let $K$ be a compact of the plane of Lebesgues measure 0 and $\Omega$ a domain containing $K$. Denote by $E$ the vector space of functions that are holomorphic on $\Omega - K$.
I'm interested in ...
- 377
4
votes
2
answers
566
views
Which test functions are the divergence of a vector field?
The following apparently elementary question came out of a somewhat naive attempt to
prove that every distribution $u\in \mathscr D'(\mathbb R^2)$ with $\partial_1 u=\partial_2 u =0$ is a constant ...
- 16.5k
0
votes
1
answer
337
views
Integral inequality
Let $X$ be the d-dimensional hypercube $X=[0,1]^d$ and let $f$ and $g$ be such that $f(x) = 1$ if $x \in A$ and $0$ otherwise, $g(x)=1$ if $x \in B$ and $0$ otherwise, where $A$ and $B$ are generic ...
- 295
1
vote
0
answers
97
views
Does a (NOT necessarily positive) current have a decomposition formula?
It is well-known that for any positive (1,1)-current $T$, there is a decomposition formula according to [Siu74]. That is, $T$ can be written as an infinite sum of prime divisors plus an extra part. In ...
- 393
10
votes
1
answer
432
views
Modern version of an inequality of R. M. Gabriel for contour integrals
I am currently reading the 1998 article Dynamics of the Binary Euclidean Algorithm:
Functional Analysis and Operators by Brigitte Vallée, which cites a 1928 article by R. M. Gabriel for the following ...
- 6,206
0
votes
4
answers
376
views
A question that arises in trying to make mathematically precise a well known informal statement about analytic functions
It is often stated that a single-valued analytic function f(z) is uniquely and
completely determined if (1) it is analytic at all points of a convergent sequence of
points in the complex plane and at ...
- 6,215
0
votes
1
answer
229
views
Complemented subspaces of $\ell_p(I)$ for uncountable $I$
I was looking for an article mimicing result of Pelczynski for $\ell_p$. I have found this one
Rodriguez-Salinas, B. (1994). On the Complemented Subspaces of $c_0(I)$ and $\ell_p(I)$ for $1 < p &...
- 1,697
3
votes
1
answer
199
views
Is P(X) a connected set for a set X with a $\sigma$-algebra P(X) and a measure function m on it to [0,$\infty$] when P(X) is equiped with meter d, that for every A,B in P(X), $d(A,B)=m(A \Delta B)$?
look at Problem14.12 of chapter3 of "Aliprantis-Burkinshaw-Principles of real analysis-3ed.1998" ; 12. Let A be the collection of all measurable subsets of X of finite measure. That is, A = {B in X: m(...
2
votes
1
answer
1k
views
Weierstrass factorization theorem in several variables
Can one indicate to me the Weierstrass factorization theorem in several variables (real or complex). In one complex variable the result is well known. Thank you in advance.
- 1,197
9
votes
3
answers
4k
views
Projections in Banach spaces
Dear All,
I am absolutely lost in the following problem:
Let $P_s, \: s \in [0,1],$ be a uniformly bounded family of projections (idempotents) in a Banach space $X$ such that $P_s P_t = P_{{\rm min}...
- 93
1
vote
1
answer
368
views
Is Every Symmetric Operator on the Schwartz Space Essentially Self-Adjoint?
More generally, suppose $S$ is a subspace of a Hilbert space $H$ that contains an orthonormal basis of $H$ (For example- the Schwartz space inside $L^2(\mathbb{R}^n)$). If $A:S \rightarrow S$ is ...
- 922
11
votes
1
answer
368
views
Sets with zeta functions that are not the primes
Does there exist a set $S \subset \mathbb N$ such that the Dirchlet density of $S$ is well-defined and positive, the Dirchlet density of $S \cap \operatorname{PRIMES}$ is well-defined and zero, and:
$...
- 149k
0
votes
1
answer
403
views
is the limit of ergodic functions still ergodic?
under what conditions is the limit of a sequence of ergodic functions still ergodic? are there simple counter-examples to this general statement?
- 21
0
votes
0
answers
231
views
Pure greedy algorithm
I study pure greedy algorithms in different basises. I am interested in 1 one question: is there such a Riesz basis $D$ in Hilbert space and $f\in H$ such that
$\|f-G_m(f,D)\|>Cm^{-1/2}\lvert\{f}\...
5
votes
2
answers
472
views
Complex structures on $R^{2N}$ with complex annulus
Let $M$ be a complex manifold of dimension $N\ge2$ such that
$\qquad$(1) $M$ is diffeomorphic to $R^{2N}$,
$\qquad$(2) There is a compact set $K\subseteq M$ such that $M\setminus K$ is biholomorphic ...
- 53
1
vote
0
answers
358
views
an infinite series expansion in terms of the polylogarithm function
we have the complex valued function :
$$f(z)=\sum_{n=0}^{\infty}a_{n}Li_{-n}(z)$$
we wish to recover the coefficients $a_{n}$ . the only thing i though would work is to try and come up with a function ...
- 181
1
vote
0
answers
289
views
Inequality regarding $\ell_p$ norms, $p<1$
Let $(x_{i,j})$ be an infinite double sequence of nonnegative real numbers, and $ 0< p<1$.
I would like to know whether one can bound from above the sum
\begin{equation}
\sum_{i,j} x_{i,j}^p
\...
- 61
1
vote
1
answer
393
views
The Dirichlet series of the Hasse–Weil L-function
I have the following question:
Is there is a paper claiming that the Dirichlet series of the Hasse–Weil $L$-function (associated with an elliptic curve over rationals) is of finite order.
Thank you in ...
- 1,197
1
vote
1
answer
298
views
Maximal spectrum of a complex, unital and commutative Banach-algebra
Let $A$ be a complex, unital and commutative Banach-algebra.
Question: Is the maximal spectrum $Max(A)$ of $A$ endowed with the topology induced by the prime spectrum $Spec(A)$ of $A$, Hausdorff?
...
- 328
4
votes
0
answers
140
views
When is $A^*A$ invertible for Banach space?
Let's consider a linear functional $A$ from smooth objects to smooth ones. It is first order operator in the sense that it extends to be a map from $W^{k+1,p}$ to $W^{k,p}$. Assume that we have $L^2$ ...
- 527
20
votes
1
answer
1k
views
Provable zero-free region for any entire function that analytically is similar to zeta(s)
Is there an entire function $f:\mathbb C\rightarrow\mathbb C$ such that for some $\delta>0$:
$f(z)$ is bounded when $\Re z>1+\delta$
$f(z)$ is unbounded when $\Re z=1$
$f(z)$ grows polynomially ...
- 1,243
0
votes
2
answers
160
views
Bounded inverse to morphism of Banach algebras
Let $A:X\to Y$ be a surjective morphism of Banach spaces.
1) Does there always exists $B_R$, a bounded right inverse to $A$?
2) Assume additionally that $A$ is a morphism of unital Banach algebras. ...
- 580
2
votes
1
answer
132
views
Form of finite dimensional contractive projection in $L_p$
Let $P$ be a finite dimensional contractive (norm 1) projection in $L_p$, $1 < p < \infty$. Then $P$ is of the following form:
$Pf = \sum_{k=1}^n g_k \int h_kf$
Where $\|g_k\|_p = \|h_k\|_q = \...
- 311
8
votes
2
answers
464
views
Direct proof of "K is projective iff C(K) has the Hahn-Banach property" ?
An object $X$ of a given category is called projective if for each morphism $f : X \rightarrow Z$, and each epimorphism $ g : Y \twoheadrightarrow Z$, there is a morphism $h : X \rightarrow Y$ such ...
- 7,249
6
votes
6
answers
1k
views
Proving continuity on spaces of distributions?
Let $\mathcal{D}'(\Omega)$ be the space of distributions on an open set $\Omega$, and $\mathcal{E}'(\Omega)$ the compactly supported ones.
When you have a linear operator $T:\mathcal{D}'(\Omega)\...
- 61
1
vote
1
answer
142
views
Linear Maps between $L^1$-spaces of singular measures
I posted the following question also here, but thought that I can get more answers in MO.
Let $(\Omega,\Sigma)$ be a measurable space and $\nu_1$, $\nu_2$ two probability measures on it. For $i=1,2$, ...
- 13
0
votes
1
answer
335
views
Proof that Euler's function cannot be continued beyond the open disc?
It is claimed on it's Wikipedia page that Euler's function, defined by the infinite product $\prod_1^\infty(1-q^n)$ for $|q|<1$, cannot be analytically continued outside the unit disc, that is, the ...
- 2,480
4
votes
1
answer
535
views
A converse of the maximum modulus Theorem
W. Rudin in Real and Complex Analysis (262) mentioned that
Theorem Suppose $M$ is a vector space of continuous complex functions
on the closed unit disc $\bar U$, with
the following properties:
(a) $...
- 43
4
votes
1
answer
615
views
Isometric embeddings of $\ell_q^m$ into $\ell_p$ and $L_p$ for $p,q\in[1,+\infty]$
I'm looking for articles describing or proving nonexistence of isometric embeddings of $m$-dimensional space $\ell_q^m$ into $L_p$ and $\ell_p$ for $q,p\in[1,+\infty]$.
Since $\ell_q^m$ is finite ...
- 1,697
6
votes
0
answers
369
views
Paving conjecture for Toeplitz matrices
Let me first recall what is the so-called paving conjecture:
for any $\epsilon >0$, there exists $r\in \mathbb N$ such that
for any bounded operator $A$ on $\ell^2(\mathbb Z)$, there exists a ...
- 16.2k
2
votes
2
answers
2k
views
Does the Fourier series of an $L^1$ function converge to the function *weakly* in $L^1$?
Let $f$ be a periodic $L^1$ function, and $S_n[f]$ the $n$-th partial sum of its Fourier series. I am aware that $S_n[f]$ might not converge toward $f$ in $L^1$ (i.e., in norm). However, does it at ...
- 32.5k
12
votes
4
answers
2k
views
Interesting results for open Riemann surfaces
As far as I know, interesting results for open Riemann surfaces are quite rare. One of them is the theorem of Gunning and Narasimhan, which asserts that every connected open Riemann surface admits a ...
- 643
0
votes
0
answers
244
views
Checking whether this would be bounded
It may be better to post this question here. Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric
positive-semi-definite matrix with exactly $m/2$ positive eigenvalues
and every entry of ...
- 1
2
votes
1
answer
637
views
Partial order on self-adjoint extensions?
Is there a natural partial order and/or lattice structure on the set of closed symmetric or self-adjoint extensions of a densely defined, unbounded, symmetric operator on a Hilbert space? Any ...
- 21.6k
3
votes
2
answers
2k
views
Dual space pairing question (Sobolev space, Bochner space)
Let $f \in H^{-1}(U)$ and $u \in H^1(U)$. I know that we write $f(u)$ as the pairing $$\langle f, u \rangle_{H^{-1}, H^1}$$.
Suppose that $v$ is the weak/distributional derivative of $u$. So
$$\...
- 55
8
votes
1
answer
847
views
A doubt about the parts of the spectrum of tensor products
Let $\mathcal{H}$ be any complex Hilbert space of infinite dimensional. By an operator $T$ I mean a linear bounded transformation from $\mathcal{H}$ into $\mathcal{H}$, i.e, $T:\mathcal{H}\rightarrow\...
- 81
9
votes
4
answers
2k
views
Books about capacity theory
While I was studying the book Variation et Optimisation de formes by Antoine Henrot and Michel Pierre, I encountered a section about the capacity associated to the $H^1$ norm, which is defined for ...
- 2,222
2
votes
1
answer
221
views
Non-perfect type one C^*-algebra, and a lemma in Fourier analysis
I would like to know if the following is true :
Let $\mathcal{H}$ be the complex Hilbert space $L^2([0,1])$ for the Lebesgue measure.
Let $q$ be the orthogonal projection on the subspace of $\mathcal{...
- 42.4k
6
votes
1
answer
526
views
Strong convergence of projections in $B(H)$
(I asked this question at math stackexchange 4 months ago, but received no answers)
Let $\{e_{kj}\}$ be the canonical matrix units in $B(H)$, with $H$ separable. Define projections $q_k$ by
$$
q_k=\...
- 2,793
1
vote
1
answer
338
views
On the generalization of the Mittag-Leffler function and fractional derivative
The Mittag-Leffler function $E_{\alpha}(x)$ has an important property:
$$
\frac{\partial^{\alpha}}{\partial x^{\alpha}} E_{\alpha}(x^{\alpha}) = E_{\alpha}(x^{\alpha}).
$$
I tried to find an ...
- 1,329
5
votes
2
answers
631
views
Proving that a complicated function is eventually concave
I have a function $f:\mathbb{R}^+ \to \mathbb{R}^+$ that I want to prove is eventually concave - i.e. that there exists $\gamma _0 > 0$ such that for every $\gamma>\gamma_0$, $f(\gamma)$ is ...
- 392
-2
votes
3
answers
850
views
Books on analytic functions on Banach spaces over a non-Archimedean field
I'm looking for good textbooks on analytic functions on Banach spaces over a non-Archimedean field.
If you know one(s), please let me know.
- 1,169
1
vote
2
answers
263
views
Books on real and/or complex analytic functions on Banach spaces taking values in Banach spaces
I'm looking for good textbooks on the subjects.
If you know one(s), please let me know.
- 1,169
2
votes
1
answer
2k
views
Monge–Ampère operator
I'm studying the article of Bedford–Taylor "Fine topology, Šilov boundary…" but I don't
understand the proof of the following proposition.
Let $u$, $v$ be plurisubharmonic functions defined ...
- 29
2
votes
1
answer
190
views
Completeness for spaces of eventually bounded nets
Let $A$ be a directed set, and $\ell^\infty_A$ the (complex vector) space of all
eventually bounded nets $A\to \mathbb{C}$. We can define the limit superior seminorm on $\ell^\infty_A$:
$$
\vert\vert{...
- 281
5
votes
2
answers
1k
views
Is this a "folk theorem" about analytic functions of a complex variable?
In a comment on question 110345 I made a claim that might be incorrect. I claimed that if
f(z) is a non-constant analytic function defined by a power series whose circle of convergence C
has a ...
- 6,215