All Questions
12,776 questions
27
votes
3
answers
948
views
A point set of power series with coefficients in {-1, 1}. Connected or not?
Let $z$ be a fixed complex number with $|z|<1$ and consider the set
$$X_z := \Big\{\sum\limits_{i=1}^{\infty} a_i z^i \ \Big|\ a_i\in \{-1,1\} \forall i\Big\}.$$
What can be said about the set $M$ ...
- 373
27
votes
4
answers
3k
views
Genealogy of the Lagrange inversion theorem
A wonderful piece of classic mathematics, well-known especially to combinatorialists and to complex analysis people, and that, in my opinion, deserves more popularity even in elementary mathematics, ...
- 60.5k
27
votes
2
answers
8k
views
Compact embeddings of Sobolev spaces: a counterexample showing the Rellich-Kondrachov theorem is sharp
Let $U$ be an open bounded subset of $\mathbb{R}^n$ with $C^{1}$ boundary. Let $1 \leq p < n$ and $p^{\ast} = pn/(n-p)$. Then the Sobolev space $W^{1,p}(U)$ is contained $L^{p^{\ast}}(U)$ and ...
- 309
27
votes
4
answers
8k
views
Proofs of Young's inequality for convolution
For $1\leq p,q \leq \infty$ such that $\frac1p +\frac1q\geq 1$, Young's inequality states $\|f\star g\|_r\leq \|f\|_p\|g\|_q$ (we work on $\mathbf{R}^d$ here), where $1+\frac1r = \frac1p+\frac1q$. ...
- 3,425
27
votes
3
answers
5k
views
Weak and Strong Integration of vector-valued functions
This is probably an elementary question, but outside my area of expertise, and I was unable to find any suitable reference:
Suppose $f:X\to E$ is a continuous function from a compact spaces (endowed ...
- 741
27
votes
1
answer
1k
views
Do Sobolev spaces contain nowhere differentiable functions?
Does the Sobolev space $H^1(R^n)$ of weakly differentiable functions on a bounded domain in $R^n$ (or a more general Sobolev space) contain a continuous but nowhere differentiable function?
- 3,873
27
votes
3
answers
2k
views
Kasteleyn's formula for domino tilings generalized?
It seems a marvel when a bunch of irrational numbers "conspire" to become rational, even better an integer. An elementary example is $\prod_{j=1}^n4\cos^2\left(\pi j/(2n+1)\right)=1$.
Kasteleyn's ...
- 43.2k
27
votes
2
answers
5k
views
What can be said about the Fourier transforms of characteristic functions?
What can be said about the Fourier transform of the characteristic function $1_A$, where $A\subset \mathbb{R}^n$ is of finite Lebesgue measure? In particular,
What properties are common to ...
- 2,719
27
votes
1
answer
4k
views
Criteria for boundedness of power series
Consider a power series $\sum_{n=0}^{\infty} a_n x^n$ that is convergent for all real
x, thus defining a function $f: \mathbb{R} \to \mathbb{R}$.
Can one give necessary and sufficient criteria the ...
- 4,014
27
votes
1
answer
1k
views
The dual of $\mathrm{BV}$
$\DeclareMathOperator\BV{BV}\DeclareMathOperator\SBV{SBV}$I'm going to let $\BV := \BV(\mathbb{R}^d)$ denote the space of functions of bounded variation on $\mathbb{R}^d$. My question concerns the ...
- 683
27
votes
0
answers
1k
views
Unital $C^{*}$ algebras whose all elements have path connected spectrum
A unital $C^{*}$ algebra is called a "Path connected algebra" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$.
What is an example of a non commutative ...
- 356
26
votes
6
answers
8k
views
prime ideals in C([0,1])
It is clear that each maximal ideal in ring of continuous functions over $[0,1]\subset \mathbb R$ corresponds to a point and vice-versa.
So, for each ideal $I$ define $Z(I) =\{x\in [0,1]\,|\,f(x)=0, ...
- 5,063
26
votes
3
answers
4k
views
Riemann mapping theorem for homeomorphisms
How do you prove to any two simply-connected domains in the plane are homeomorphic without using the Riemann mapping theorem? An elementary proof would be nice.
- 1,159
26
votes
3
answers
16k
views
the dual space of C(X) (X is noncompact metric space)
It is well known that when $X$ is a compact space (or locally compact space), the dual space of $C(X)=\{f |f:
X\rightarrow \mathbb{C} \text{ is continuous and bounded} \}$ is $M(X)$, the space of ...
- 1,706
26
votes
2
answers
5k
views
Does Arzelà-Ascoli require choice?
Inspired by a recent Math.SE question entitled Where do we need the axiom of choice in Riemannian geometry?, I was thinking of the Arzelà--Ascoli theorem. Let's state a very simple version:
...
- 29.8k
26
votes
4
answers
5k
views
Can $L^{2}$ be represented as a space of functions (not equivalence classes)?
Let $X$ be the vector space of all Lebesgue-measurable functions $f:\left[a,b\right]\rightarrowℝ$ such that $\int^{b}_{a}\left|f\left(x\right)\right|^{2}dx<\infty$ (Lebesgue integral). Then we can ...
- 4,597
26
votes
2
answers
6k
views
Understanding a simplifying assumption in proof of the invariant subspace problem
In a recent preprint On the invariant subspace problem in Hilbert spaces Per H. Enflo claims to have solved the invariant subspace problem, showing that every bounded linear operator on a separable ...
- 423
26
votes
3
answers
2k
views
About the category of von neumann algebras
I am looking for one (or more) reference about properties of the category of von Neumann algebra.
More precisely, in an answer of a previous question, Dmitri Pavlov mentions
that the $W^*$ category ...
- 357
26
votes
3
answers
7k
views
Dual of bounded uniformly continuous functions
Let $(X,d)$ be a metric space, and let $C_u(X)$ be the Banach space of bounded uniformly continuous functions on $X$ (with the uniform norm). How can I characterize its dual space $C_u(X)^*$?
I ...
- 29.8k
26
votes
2
answers
3k
views
Corollaries of the Yoneda Lemma in Analysis?
This is a cross-post of my ~2 weeks (canonically) unanswered question on Math.SE: https://math.stackexchange.com/questions/1830287/corollaries-of-the-yoneda-lemma-in-analysis.
I am looking for some ...
- 2,680
26
votes
2
answers
1k
views
Origin and first uses of $\ell_p$ norms?
When exactly were $\ell_p$ norms first defined and used?
(Here is what I know, or think I know: Lebesgue and/or Riesz had something to do with them, but in some sense they go back to Minkowski, since ...
- 20.2k
26
votes
3
answers
2k
views
Universality of zeta- and L-functions
Voronin´s Universality Theorem (for the Riemann zeta-Function) according to Wikipedia: Let $U$ be a compact subset of the "critical half-strip" $\{s\in\mathbb{C}:\frac{1}{2}<Re(s)<1\}$ with ...
- 7,127
26
votes
1
answer
820
views
The maximal "nearly convex" function
The following problem is only tangentially related to my present work, and I do
not have any applications. However, I am curious to know the solution -- or
even to see a lack thereof, indicating that ...
- 23k
26
votes
2
answers
1k
views
Symmetric strengthening of the Cauchy-Schwarz inequality
In this great question by Nathaniel Johnston, and in its answers, we can learn the following remarkable inequality: For all $v,w \in \mathbb{R}^n$ we have
\begin{align*}
\|v^2\| \, \|w^2\| - \langle ...
- 12.6k
26
votes
2
answers
2k
views
When is a locally convex topological vector space normal or paracompact?
All locally convex topological vector spaces (LCTVS) are completely regular, since their topology is given by a family of semi-norms. I'm interested in conditions that imply that a LCTVS is ...
- 26.8k
26
votes
1
answer
1k
views
Why are quasitopological spaces needed in sheaf theoretic approaches to the h-principle?
Recently I have been learning more about the h-principle and in particular the methods of "continuous sheaves". In many treatments of this I see people using "quasi-topological spaces" and I am trying ...
- 27.5k
26
votes
3
answers
11k
views
L1 distance between gaussian measures
L1 distance between gaussian measures: Definition
Let $P_1$ and $P_0$ be two gaussian measures on $\mathbb{R}^p$ with respective "mean,Variance" $m_1,C_1$ and $m_0,C_0$ (I assume matrices have full ...
- 833
25
votes
16
answers
4k
views
functions satisfying "one-one iff onto"
Hello Everybody.
I need some more examples for the following really interesting phenomenon:
A function from the class ... is one-one iff it is onto.
Some ...
25
votes
6
answers
15k
views
Does every distribution define a Radon measure?
On the one hand, Wikipedia suggests that every distribution defines a Radon measure:
http://en.wikipedia.org/wiki/Distribution_(mathematics)#Functions_as_distributions (revision from February 2010, ...
- 2,895
25
votes
2
answers
4k
views
Understanding of rough path
A rough path is defined as an ordered pair
$ (X, \mathbb X)$, where $X$ is a path mapping from $[0,T]$ to some Banach space $V$
and $\mathbb X:[0,T]^2 \mapsto V^2$ is another mapping for additional ...
- 1,399
25
votes
2
answers
4k
views
Dual of the space of Hölder continuous functions?
Let $X=C^{\alpha}(\Omega,\mathbb{R})$ be the space of Hölder continuous functions. What is its dual?
- 1,256
25
votes
5
answers
6k
views
When I can safely assume that a function is a Laplace transform of other function?
If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is $\hat{f}(s)$ such that my function $f(x)$ can be written as:
$$f(x) =...
- 923
25
votes
1
answer
3k
views
Does there exist a measurable function which is not a.e. "strongly" measurable?
More specifically, letting $I=[0,1]$, do there exist $f,E$ with $E$ a (necessarily nonseparable) Banach space and $f$ a bounded Lebesgue measurable function $I\to E$ such that $f$ is not equal almost ...
- 3,584
25
votes
1
answer
2k
views
Can we just use the linear term of exponential sums to sum divergent series
Suppose you want to compute the sum $\sum_{n=0}^{\infty} a_n $
You could consider the expression $f(x) = \sum_{n=0}^{\infty} e^{a_n x}$ and try to compute the coefficient of an $x^1$ term in the ...
- 2,689
25
votes
2
answers
2k
views
$f^3,f^2$ are the cube and quadratic of f respectively and both infinite differentiable on $R$,how to show so is $f$
Let $f$ be a real function with domain R.
If $f^2$ and $f^3$ are both infinitely differentiable on R,
how to prove $f$ is infinitely differentiable on R?
I have been thinking about this problem for a ...
- 295
25
votes
2
answers
1k
views
Can nuclearity be determined by tensoring with a single C*-algebra?
A C*-algebra is nuclear if the algebraic tensor product $A\odot B$ ($B$ is any other C*-algebra) admits a unique C*-norm. This definition requires testing the condition for nuclearity with `all' C*-...
- 505
25
votes
6
answers
3k
views
Quantum fields and infinite tensor products
As I understand it, a naive interpretation of the state space of a quantum field theory is an infinite tensor product
$$\otimes_{x\in M} H_x,$$
where $x$ runs over the points of space. This ...
- 13.6k
25
votes
2
answers
2k
views
Functional approach vs jet approach to Lagrangian field theory
Context: I am a PhD student in theoretical physics with higher-than-average education on differential geometry. I am trying to understand Lagrangian and Hamiltonian field theories and related concepts ...
- 3,292
25
votes
3
answers
13k
views
Fourier transform of the unit sphere
The Fourier transform of the volume form of the (n-1)-sphere in $\mathbf R^n$ is given by the well-known formula
$$
\int_{S^{n-1}}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u) = (2\pi)^{\nu ...
- 31.5k
25
votes
1
answer
8k
views
Convergence of Fourier Series of $L^1$ Functions
I recently learned of the result by Carleson and Hunt (1968) which states that if $f \in L^p$ for $p > 1$, then the Fourier series of $f$ converges to $f$ pointwise-a.e. Also, Wikipedia informs me ...
- 871
25
votes
1
answer
1k
views
When does a modular form satisfy a differential equation with rational coefficients?
Given a modular form $f$ of weight $k$ for a congruence subgroup $\Gamma$, and a modular function $t$ with $t(i\infty)=0$, we can form a function $F$ such that $F(t(z))=f(z)$ (at least locally), and ...
- 4,593
25
votes
1
answer
1k
views
Comparison map between de Rham cohomology of analytic and formal neighborhoods of singularities
Suppose that $X$ is a complex algebraic (or complex analytic) variety, and $x \in X$ is a singular point. I am interested in two types of local differential forms at $x$: analytic and formal.
First, ...
- 536
25
votes
1
answer
782
views
Decidability of equality of elementary expressions
In the following definition the term expression is to be understood as a finite tree built from formal symbols without any predefined meaning assigned to them.
Define the set $\mathcal{E}$ of ...
- 6,819
24
votes
12
answers
4k
views
2D problems which are easier to solve in 3D
It sometimes happens that 1D problems are easier to solve by somehow adding a dimension. For example, we convert linear differential equations for a real unknown to a complex unknown (to use complex ...
24
votes
4
answers
3k
views
Why the sequence of Bernstein polynomials of $\sqrt x$ is increasing?
Bernstein polynomials preserves nicely several global properties of the function to be approximated: if e.g. $f:[0,1]\to\mathbb R$ is non-negative, or monotone, or convex; or if it has, say, non-...
- 60.5k
24
votes
3
answers
3k
views
How to find a conformal map of the unit disk on a given simply-connected domain
By the classical Riemann Theorem, each bounded simply-connected domain in the complex plane is the image of the unit disk under a conformal transformation, which can be illustrated drawing images of ...
- 41.9k
24
votes
2
answers
2k
views
Is the Invariant Subspace Problem arithmetic?
Invariant Subspace Conjecture: A bounded operator on a separable Hilbert space has a non-trivial closed invariant subspace.
Can this conjecture be reformulated as an arithmetic statement, that is, $\...
- 6,891
24
votes
3
answers
4k
views
Self-dual normed spaces which are not Hilbert spaces
Are there any examples of non-Hilbert normed spaces which are isomorphic (in the norm sense) to their dual spaces? Or, is there any result in Functional Analysis which says that if a space is self-...
- 2,239
24
votes
3
answers
9k
views
Functions of several complex variables: book recommendations?
Can anyone recommend a good comprehensive introduction to functions of several complex variables that a) is fairly up to date, b) isn't a geometry or an algebra book only, but takes multiple ...
24
votes
2
answers
2k
views
Unique predual of a Banach space
Suppose $E$ is a dual Banach space whose predual is unique, and $E_0$ is a codimension 1 weak* closed subspace of $E$. Is the predual of $E_0$ necessarily unique?
Okay, I will reveal the motivation. ...
- 42.8k