All Questions
10,049 questions
22
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1
answer
4k
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Image of the trace operator
It is well-known that we have the trace theorem for Sobolev spaces. Let $\Omega$ be an open domain with smooth boundary, we know that the map
$$ T: C^1(\bar\Omega) \to C^1(\partial\Omega) \subset L^...
22
votes
1
answer
3k
views
Reference for Diagonalization Trick
There is a standard trick in analysis, where one chooses a subsequence, then a subsequence of that... and wants to get an eventual subsubsequence of all of them and you take the diagonal. I've always ...
22
votes
2
answers
652
views
Does every positive continuous function have a non-negative interpolating polynomial of every degree?
Let $f:[a,b] \to (0,\infty)$ be a continuous function. Then is it necessarily true that for every $n\ge 1$, we can find $n+1$ distinct points $\{x_0,x_1,...,x_n\}$ in $[a,b]$ such that the ...
22
votes
1
answer
745
views
The Mackey Topology on a Von Neumann Algebra
Every von Neumann algebra $\mathcal M$ is the dual of a unique Banach space $\mathcal M_* $. The Mackey topology on $\mathcal M$ is the topology of uniform convergence on weakly compact subsets of $\...
22
votes
2
answers
2k
views
Examples of loss of regularity by "creation of topology"
I would like to have a list as general as possible of examples of situations where the density of smooth objects into some "natural class" (the meaning of "natural" depending on the problem considered)...
21
votes
8
answers
11k
views
Nice applications of the spectral theorem?
Most books and courses on linear algebra or functional analysis present at least one version of the spectral theorem (either in finite or infinite dimension) and emphasize its importance to many ...
21
votes
5
answers
4k
views
Isomorphisms of Banach Spaces
Suppose $X$ and $Y$ are Banach spaces whose dual spaces are isometrically isomorphic. It is certainly true that $X$ and $Y$ need not be isometrically isomorphic, but must it be true that there is a ...
21
votes
4
answers
2k
views
Why are currents named currents?
Why do currents, functionals on compactly supported differentiable n-forms, bear the name they do?
I've assumed that it has something to do with an electrical current being formalized as a vector ...
21
votes
7
answers
2k
views
Identities and inequalities in analysis and probability
Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...
21
votes
5
answers
18k
views
When is Sobolev space a subset of the continuous functions?
If we let $\Omega\subset\mathbb{R}^d$ with $d=1,2,3$ and define $\mathcal{H}^1(\Omega)=(w\in L_2(\Omega): \frac{\partial w}{\partial x_i}\in L_2(\Omega), i=1,...,d)$. My tutor has repeated several ...
21
votes
1
answer
3k
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The list of problems for Grothendieck's thesis
Is the list of open problems which were given by Dieudonne and Schwartz to Grothendieck for his thesis published somewhere? I know a quotation of Dieudonne that the problems concerned duality theory ...
21
votes
3
answers
3k
views
Can you tell whether a space is Banach from the unit ball?
Let $V$ be a real vector space. It is well known that a subset $B\subset V$ is the unit ball for some norm on $V$ if and only if $B$ satisfies the following conditions:
$B$ is convex, i.e. if $v,w\...
21
votes
2
answers
3k
views
A measure on the space of probability measures
This question was originaly posted in the stackexchange https://math.stackexchange.com/questions/1226701/a-measure-on-the-space-of-probability-measures but since it only got a comment I decided to ...
21
votes
4
answers
2k
views
A question on the integral of Hilbert valued functions
This questions stems from an attempt to recast in a form suitable for teaching some standard computations which are usually proved by handwaving, without much care about the details. My hope is that ...
21
votes
1
answer
1k
views
Is Dependent Choice all we really need?
http://en.wikipedia.org/wiki/Axiom_of_dependent_choice
Is DC sufficient for the understanding of objects that are countable in some suitable sense?
For example, is DC sufficient for the full ...
21
votes
1
answer
742
views
Non real eigenvalues for elliptic equations
I am looking for an example of a pure second order uniformly elliptic operator
$L=\sum_{i,j=1}^da_{ij}(x)D_{ij}$ in a bounded domain $\Omega$ (with Dirichlet boundary conditions, for example) having a ...
21
votes
2
answers
1k
views
Is there an L^p tauberian theorem?
From Wiener's tauberian theorem we know that linear combinations of translates of f \in L^1(R) are dense in L^1(R) if and only if the Fourier transform of f never vanishes. It is also known that ...
21
votes
1
answer
690
views
Diameter of a quotient of the infinite dimensional sphere
Suppose a group $\Gamma$ acts by isometries on the Hilbert space $\mathbb{H}^\infty$ and it fixes the origin. So $\Gamma$ acts on the unit sphere $\mathbb{S}^\infty$ as well.
Assume that the action $...
21
votes
2
answers
1k
views
Closed subspaces of Banach spaces
Is it true that, assuming the Axiom of Choice, every infinite-dimensional Banach space has an infinite-dimensional closed subspace with infinite codimension? Note that this is different from the ...
21
votes
1
answer
2k
views
Almost commuting unitary matrices
Suppose that $A_1,\dots, A_k$ are unitary matrices such that any two of them can be approximated by commuting unitary matrices. i.e. for any $i$ and $j$, there are unitary matrices $A_i'$ and $A_j'$ ...
21
votes
3
answers
1k
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Separating pure states on the $2\times 2$ matrix algebra
I have an idea for a possible counterexample to the noncommutative Stone-Weierstrass problem. A good answer to the following question would really help.
Let $\mathcal{A}$ be the C*-algebra of $2\...
21
votes
1
answer
835
views
On complemented von Neumann algebras
Edit: according to Narutaka Ozawa, question 3) is still open in the type $\mathrm{II}_1$ case. In other terms, it is not known whether every topologically complemented type $\mathrm{II}_1$ factor in $...
21
votes
1
answer
3k
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Density of polynomials in $C^k(\overline\Omega)$
Let $\Omega$ be an open and bounded subset of $\mathbb{R}^2$ and let $C^k(\Omega)$, $1\leq k<\infty$, be the space of functions $f$ with continuous derivatives of order $\leq k$ in $\Omega$, ...
21
votes
1
answer
939
views
Is Grothendieck classification of tensor norms and Kuratowski's 14 sets theorem somehow related?
It is known that there are only 14 reasonable tensor norms in $Ban$. On the other hand it is well known fact for topologists that one can obtain only 14 different sets from a given set applying ...
21
votes
2
answers
2k
views
In a Banach algebra, do ab and ba have almost the same exponential spectrum?
Let $A$ be a complex Banach algebra with identity 1. Define the exponential spectrum $e(x)$ of an element $x\in A$ by $$e(x)= \{\lambda\in\mathbb{C}: x-\lambda1 \notin G_1(A)\},$$ where $G_1(A)$ is ...
21
votes
1
answer
1k
views
"Minimal" group C*-algebra?
Let $\Gamma$ be a discrete group (though this could be asked for general locally compact groups) and consider the Banach $*$-algebra $\ell^1(\Gamma)$. We have two natural $C^*$-algebra completions: ...
21
votes
2
answers
2k
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Uncertainty principle and Cramer-Rao bound - is there relation?
Just out of curiosity.
The two things sounds a little bit similar - 1) Uncertainty principle 2) Cramer-Rao bound.
Saying that we cannot measure something with certain accuracy.
However looking closer ...
21
votes
2
answers
1k
views
Meager subspaces of a Banach space and weak-* convergence
I previously asked a version of this question on Math.SE, but didn't receive an answer. (But there is a bounty there if you want to claim it!)
Let $X$ be a Banach space. (If it helps, feel free to ...
21
votes
2
answers
2k
views
A strange variant of the Gaussian log-Sobolev inequality
Let $\phi : \mathbb{R}^d \to \mathbb{R}$ be a convex function, and assume that it grows at most linearly at infinity for simplicity. Denote by $\gamma$ the standard Gaussian measure on $\mathbb{R}^d$, ...
21
votes
0
answers
869
views
Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials
While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...
21
votes
0
answers
732
views
Closed connected additive subgroups of the Hilbert space
It is a classical result that a closed and connected additive subgroup of $\mathbb{R}^n$ is necessarily a linear subspace. However, this is no longer true in infinite dimension: a very easy example is ...
21
votes
0
answers
876
views
Are the eigenvalues of the Laplacian of a generic Kähler metric simple?
It is a theorem of Uhlenbeck that for a generic Riemannian metric, the Laplacian acting on functions has simple eigenvalues, i.e., all the eigenspaces are 1-dimensional. (Here "generic" means the set ...
20
votes
12
answers
9k
views
The role of completeness in Hilbert Spaces
Why do Hilbert spaces have to be complete?
I've been studying (teaching myself about) Hilbert spaces for a while now as they have a habit of popping up in many of the papers I'm come across (I'm a ...
20
votes
7
answers
5k
views
Why do infinite-dimensional vector spaces usually have additional structure?
On Mathematics Stack Exchange, I asked the following question: Why are infinite-dimensional vector spaces usually equipped with additional structure? Although it received one good answer, I feel that ...
20
votes
8
answers
12k
views
Can a self-adjoint operator have a continuous set of eigenvalues?
This should be a trivial question for mathematicians but not for typical physicists.
I know that the spectrum of a linear operator on a Banach space splits into the so-called "point," "continuous" ...
20
votes
3
answers
4k
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What is the origin of the term "spectrum" in mathematics?
The use of the term "spectrum" to denote the prime ideals of a ring originates from the case that the ring is, say, $\mathbb{C}[T]$ where $T$ is a linear operator on a finite-dimensional vector space; ...
20
votes
6
answers
7k
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Does the derivative of log have a Dirac delta term?
Dirac writes down the following formula on page 61 of his "Principles of quantum mechanics":
$\frac{d}{dx}\log x = \frac{1}{x} -i\pi\delta(x)$, see http://adsabs.harvard.edu/abs/1947pqm..book.....D ...
20
votes
3
answers
3k
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Realizing universal $C^*$-algebras as concrete $C^*$-algebras
How do I in general realize a universal C*-algebra generated by some generators and relation as concrete C*-algebras? For example, I know that universal C*-algebra generated by a single unitary is $C(\...
20
votes
2
answers
3k
views
Non-differentiable Lipschitz functions
As far as I understand, there are Lipschitz functions $f:\mathbb{R}\to\ell^\infty$ that are nowhere differentiable in the Frechet sense. Where can I find such an example?
20
votes
3
answers
4k
views
Basis of l^infinity
Is it possible to exhibit a (Hamel) basis for the vector space l^infinity, given by the bounded sequences of real numbers?
20
votes
2
answers
1k
views
P-adic C* algebras
I understand that there is a definition of p-adic Banach algebras and that a significant amount of functional analysis can be developed in the non-archimedean setting. Is there a p-adic version of C*-...
20
votes
2
answers
4k
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Ideals of the ring of smooth functions
The ring $C^\infty(M)$ of smooth functions on a smooth manifold $M$ is a topological ring with respect to the Whitney topology and the usual ring operations. Is it possible to describe, maybe under ...
20
votes
3
answers
8k
views
Why do inner products require conjugation?
For Hermitian matrices and operators, the most "natural" inner product is $f^H \cdot g$ or $\int f^* g\; dx$. A similar situation holds interpreting Fourier transforms as the inner product of ...
20
votes
2
answers
1k
views
Can There be a 1 dimensional Banach-Tarski paradox in the absence of choice
Let $\mathbb{R}$ act on itself by translation. Then there is no finite decomposition of a unit interval into pieces which, when translated, yields two distinct unit intervals.
More formally does ...
20
votes
2
answers
7k
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Question about functional derivatives
This page on Wikipedia defines the so-called functional derivative as follows: "Given a manifold $M$ representing (continuous/smooth) functions $\rho$ (with certain boundary conditions, etc.) and a ...
20
votes
2
answers
870
views
C$^*$-algebras isomorphic after tensoring with $M_n(\mathbb C)$
In 1977, Joan Plastiras gave a striking example of two non $*$-isomorphic C$^*$-algebras $\mathcal A$ and $\mathcal B$ such that $$\mathcal A \otimes M_2(\mathbb C) \simeq \mathcal B\otimes M_2(\...
20
votes
2
answers
1k
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The Gelfand duality for pro-$C^*$-algebras
The Gelfand duality says that
$$X\to C(X)$$
is a contravariant equivalence between the category of compact Hausdorff spaces and continuous maps and the category of commutative unital $C^*$-algebras ...
20
votes
1
answer
993
views
Which spaces are characterized by functions with compact support ?
It's well known that two locally compact Hausdorff spaces $X, Y$ are homeomorphic iff the rings $C_0(X), C_0(Y)$ (continuous functions vanishing at infinity) are isomorphic.
Is there a class $\...
20
votes
2
answers
922
views
A functional inequality about log-concave functions
Let $f,g$ be smooth even log-concave functions on $\mathbb{R}^{n}$, i.e.,$f=e^{-F(x)}, g=e^{-G(x)}$ for some even convex functions $F(x),G(x)$. Is it true that:
$$
\int_{\mathbb{R}^{n}} \langle \...
20
votes
0
answers
333
views
Existence of orthonormal basis for $L^2(G)$ in $C_c(G)$
Remark: I cross-posted this question on MSE and added a bounty to it.
Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly ...