All Questions
10,049 questions
20
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Can the similarity between the Riesz representation theorem and the Yoneda embedding lemma be given a formal undergirding?
For example, by viewing Hilbert spaces as enriched categories in some fashion? (I suppose the same idea of considering the inner product of a Hilbert space as a generalized Hom-set has also been ...
- 5,827
19
votes
4
answers
3k
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Strange result about convexity
$f \in C^2([0,1])$ with $f''$ convex and $f(0) = f'(0) = f''(0) = 0$.
Is it true that : $f''(1)+6f(1)\geq 4f'(1)$ ?
Source: AoPS
- 4,074
19
votes
6
answers
8k
views
Unbounded operator bounded in a dense subset
Let $X, Y$ be normed vector spaces, where $X$ is infinite dimensional. Does there exist a linear map $T : X \rightarrow Y$ and a subset $D$ of $X$ such that $D$ is dense in $X$, $T$ is bounded in $D$ (...
- 783
19
votes
2
answers
2k
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Can we take a supremum over all Hilbert spaces?
In my paper On the optimal error bound for the first step in the method of cyclic alternating projections, I defined functions $f_n:[0,1]\to\mathbb{R}$,
$n\geqslant 2$, by
$$
f_n(c)=\sup\{\|P_n\dotsm ...
- 935
19
votes
5
answers
16k
views
What does "kernel" mean in integral kernel?
In functional analysis, there is the term "integral kernel". Examples are Possion kernel, Dirichlet kernel etc.
In algebra, the term kernel of a homomorphism refers to the inverse image of the zero ...
user2529
19
votes
1
answer
5k
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A Fourier-analytic inequality used by Jean Bourgain
I am currently reading Jean Bourgain's 1986 paper A Szemerédi type theorem for sets of positive density in $R^k$ and would appreciate some help in understanding a Fourier-analytic estimate used in ...
- 6,206
19
votes
7
answers
2k
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Generalizations of "standard" calculus
We have the usual analogy between infinitesimal calculus (integrals and derivatives) and finite calculus (sums and forward differences), and also the generalization of infinitesimal calculus to ...
- 6,792
19
votes
3
answers
1k
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What standard Banach space is isomorphic to the completion of this different normed structure on $\ell^1$?
A colleague asked me the following question:
"What can one do with the following norm on $\ell^1$: $|x|=\int_1^2 |x|_pdp$ where $| \;\; |_p$ is the standard norm on $\ell_p$?"
This ...
- 356
19
votes
4
answers
5k
views
Explicit extension of Lipschitz function (Kirszbraun theorem)
Kirszbraun theorem states that if $U$ is a subset of some Hilbert space $H_1$, and $H_2$ is another Hilbert space, and $f : U \to H_2$ is a Lipschitz-continuous map, then $f$ can be extended to a ...
- 1,839
19
votes
1
answer
773
views
Are algebraically isomorphic $C^*$-algebras $*$-isomorphic?
If A and B are C^*-algebras that are algebraically isomorphic to each other, does
this imply that they are *-isomorphic to each other?
- 213
19
votes
1
answer
5k
views
Intuition for the Hardy space $H^1$ on $R^n$
the standard intuition for Lebesgue spaces $L^p(\mathbb R^n)$ for $p \in [1,\infty]$ are measurable functions with certain decay properties at infinity or at the singularities.
In particular, a ...
- 5,327
19
votes
2
answers
5k
views
Is there an infinite-dimensional Banach space with a compact unit ball?
A popular pair of exercises in first courses on functional analysis prove the following theorem:
The unit ball of a Banach space $X$ is compact if and only if $X$ is finite-dimensional.
My ...
- 11.4k
19
votes
3
answers
1k
views
Is there "Schur-Weyl duality" for infinite dimensional unitary group?
To what extent does the relation between the diagonal representation of $SU(n)$ in $(\mathbb{C}^n)^{\otimes k}$ and representations of the symmetric group $S_k$ remain valid when instead of the group $...
- 965
19
votes
3
answers
711
views
Almost isometric linear maps
Say that a linear map $\varphi : B(\mathcal H) \rightarrow B(\mathcal H)$ is $\epsilon$-almost isometric if
$$ 1 - \epsilon \leq \lVert\varphi(a)\rVert \leq 1+\epsilon, \quad \forall a\in B(\mathcal H)...
- 3,984
19
votes
1
answer
3k
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Infinite convex combinations in a Banach space
Let's say that a subset $C$ of a Banach space $X$ is $\sigma$-convex if the following property holds:
For any sequence $(x_k)_{k\ge0}$ in $C$, and for
any sequence of non-negative real numbers $(\...
- 60.5k
19
votes
0
answers
552
views
Talagrand's "Creating convexity" conjecture
We say a subset $A$ of $\mathbb{R}^N$ is balanced if
\begin{equation}
x \in A, \lambda \in [-1,1] \implies \lambda x \in A.
\end{equation}
Given a subset $A$ of $\mathbb{R}^N$, we write
\begin{...
- 484
19
votes
0
answers
1k
views
Is there some way to see a Hilbert space as a C-enriched category?
The inner product of vectors in a Hilbert space has many properties in common with a hom functor. I know that one can make a projectivized Hilbert space into a metric space with the Fubini-Study ...
- 1,532
18
votes
6
answers
4k
views
What is the best place to learn about the mathematical foundations of quantum mechanics?
I'm looking for good references to learn about the mathematical foundations of quantum mechanics. By mathematical foundations, I do not mean rigorous quantum mechanics in general but the axioms behind ...
- 1,305
18
votes
3
answers
2k
views
Research topics in distribution theory
The theory of distributions is very interesting, and I have noticed that it has many applications especially with regard to PDEs. But what are the research topics in this theory? also in terms of ...
- 589
18
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4
answers
2k
views
Does "taking the dual space" stabilize?
Every book which treats dual spaces of normend spaces states that $(c_0)' = \ell^1$ and $(\ell^1)' = \ell^\infty$ and some also describe $(\ell^\infty)'$.
However, is anything known about higher ...
- 12.7k
18
votes
3
answers
2k
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What are the right categories of finite-dimensional Banach spaces?
This is inspired partly by this question, especially Tom Leinster's answer.
Let me start with some background. I apologize that this will be rather long, since I'm hoping for input from people who ...
- 11.4k
18
votes
3
answers
2k
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Poincare lemma for non-smooth differentiable forms
The Poincare lemma is almost always formulated for differential forms with smooth coefficients (or sometimes for currents that have distributional coefficients). I would like to have it for $C^k$-...
- 16.4k
18
votes
3
answers
1k
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In which sense the GNS-construction is a functor?
I asked this at mathstackexchange a week ago, without success.
I think the Gelfand–Naimark–Segal construction must be a functor in some sense, but I can't find an explicit statement anywhere. Can ...
- 7,426
18
votes
3
answers
4k
views
Formal adjoint of the covariant derivative
Let $E \to M$ be a vector bundle over some Riemannian metric $(M, g)$ and endow it with some fibre metric. Assume that covariant derivative $\nabla$ is compatible with the metric.
It is essentially ...
- 5,824
18
votes
1
answer
564
views
Is the space of Hankel operators complemented in B(H)?
Let $H$ be $\ell^2({\mathbb N})$ and let $S:H\to H$ be the unilateral forward shift, so that $S^*S=I\neq SS^*$. Then a bounded operator $T:H\to H$ is Hankel if and only if it satisfies $TS=S^*T$.
Let ...
- 25.8k
18
votes
1
answer
3k
views
Let a function f have all moments zero. What conditions force f to be identically zero?
Throughout, let $f$ be a Lebesgue measurable function (or continuous if you wish, but this is probably no easier). (Questions with distributions etc. are possible also but I want to keep things simple ...
- 1,990
18
votes
3
answers
1k
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Spectra of elements of a Banach algebra and the role played by the Hahn-Banach Theorem.
This problem was posed on Math StackExchange some time ago, but it did not garner any solutions there. I think that it is interesting enough to be posed here on Math Overflow, so here it goes.
Let $ \...
- 816
18
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1
answer
5k
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Unbounded linear operator defined on $l^2$
Let $l^2$ be a Hilbert space of infinite sequences $(z_0, z_1, \cdots)$ with finite $\sum_{i=0}^{\infty} |z_i|^2$.
Are there any simple example of unbounded linear opearator $T: l^2 \to l^2$ with $D(...
- 2,821
18
votes
4
answers
1k
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Who first used the multiplication operator version of spectral theory
This is another history question.
Hilbert phrased the spectral theorem in terms of resolutions of the identity.
While this remained the form of Stone and von Neumann, they did also have the ...
- 827
18
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1
answer
1k
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Who introduced the notion of "stability" in numerical analysis?
I am preparing a lecture course on the applications of operator theory where I intended to make some numerical analysis application. I was wondering about this question while browsing the literature I ...
- 4,729
18
votes
1
answer
3k
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How bad can the second derivative of a convex function be?
One can easily construct an example of a measurable function $f:(a,b)\to \mathbb{R}$ which satisfies the following property:
$$\label{p}\tag{P}
f\notin L^1(I),\ \mbox{for each interval}\ I\subset (a,...
- 409
18
votes
1
answer
748
views
Banach-Mazur distance between the cube and the octahedron
The Banach-Mazur distance $d(X, Y)$ between two normed spaces $X, Y$ of the same dimension is defined as $d(X, Y) = \log\inf \|T\| \cdot \|T^{-1}\|$, where the $T:X \to Y$ is a linear and invertible ...
- 243
18
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4
answers
1k
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Reference for a strong intermediate value theorem for measures
Let $\mu$ be a finite nonatomic measure on a measurable space $(X,\Sigma)$, and for simplicity assume that $\mu(X) = 1$. There is a well-known "intermediate value theorem" of Sierpiński that states ...
- 5,407
18
votes
2
answers
1k
views
compact-open topology on $B(H)$
In topology, it is common to use the compact-open topology on the set of continuous maps between two given topological spaces.
Let now $H$ be a Hilbert space and $B(H)$ the set of continuous linear ...
- 43.2k
18
votes
2
answers
776
views
What is known about the "unitary group" of a rigged Hilbert space?
Suppose that $(E,H)$ is a rigged (infinite dimensional, separable) Hilbert space, i.e. $H$ is a Hilbert space, and $E$ is a Fréchet space, equipped with a continuous linear injection $E \rightarrow H$ ...
- 556
18
votes
3
answers
1k
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Example of a space for which $V \cong Hom(V,V)$
Let $V$ be a topological linear space, and let $\operatorname{Hom}(V,V)$ be the space of continuous linear maps from $V$ back to $V$, equipped with a suitable topology.
Is there a non-trivial ...
- 8,512
18
votes
1
answer
2k
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Borel Lemma for vector-valued functions
The classical Borel Lemma states that for an arbitrary sequence $(v_n)_{n \in \mathbb{N}_0}$ of complex numbers there is a smooth function $f\colon \mathbb{R} \longrightarrow \mathbb{C}$ with Taylor ...
- 8,098
18
votes
1
answer
1k
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Commuting unitaries
Is the following true:
For every unit vectors $x_1,..., x_n$, $y_1,..., y_n$ in $\mathbb{C}^k$
there exist a Hilbert space $H$, unitary operators $U_1,...,U_n$ and $V_1,...,V_n$ in $B(H)$ and unit ...
- 4,705
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1
answer
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Is every continuous function measurable?
This question has already been asked on Math StackExchange here, but was too old to be migrated, and I think will be more appropriate to MathOverflow.
In non-Hausdorff topology it is standard to ...
- 321
18
votes
1
answer
996
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Existance of certain almost invariant functions related to amenability and piece-wise transformations
We would like very much to know the answer to the following question:
Let $\|\cdot\|$ be any norm on $\mathbb{Z}^d$ and let $W(\mathbb{Z}^d)$ be the group of all bijections of $\mathbb{Z}^d$ such ...
18
votes
2
answers
1k
views
Complex structure on $L^2(\mathbb R)$ generalizing the Hilbert transform
The Hilbert transform on the real Hilbert space $L^2(\mathbb R)$ is the singular integral operator
$$
\mathcal H(f)(x) := \frac{1}{\pi} \int_{-\infty}^\infty \frac{1}{x-y} f(y) dy.
$$
It satisfies $\...
- 43.2k
18
votes
1
answer
2k
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Equivalence of fractional Sobolev space defined through Gagliardo norm and interpolation; dependence on the domain
Let $\Gamma$ be a smooth hypersurface in $\mathbb{R}^n$. We can define the fractional Sobolev space
$$X = \left\{ u \in L^2(\Gamma) \mid |u|_X^2 := \int_\Gamma \int_\Gamma \frac{|u(x)-u(y)|^2}{|x-y|^{...
- 201
18
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1
answer
1k
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Are there non-reflexive abelian topological groups isomorphic to their second dual?
I posted the following question in a comment at
Are there non-reflexive vector spaces isomorphic to their bi-dual? and it got one upvote, but it didn't get an answer, so I'll post it as an ...
- 50.6k
18
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0
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373
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Can Rep(G) tell us whether G is discrete?
Given a locally compact group $G$, let $$\mathrm{Rep}(G)$$ be its category of unitary representations.
The objects of that category are strongly continuous unitary representations of $G$ on Hilbert ...
- 43.2k
17
votes
5
answers
7k
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A counter example to Hahn-Banach separation theorem of convex sets.
I'm trying to understand the necessity for the assumption in the Hahn-Banach theorem for one of the convex sets to have an interior point. The other way I've seen the theorem stated, one set is closed ...
- 2,641
17
votes
4
answers
959
views
What is the minimum of this quantity on $S^{n-2}\times S^{n-2}$?
My question is to find the minimum of the following expression:
$$A(x,y) = \sum_{1\leq i<j\leq n} |x_i-x_j|\ |y_i-y_j|,$$
over the set of pairs of real vectors $x=(x_1,\dots,x_n),y=(y_1,\dots,y_n)$ ...
- 4,484
17
votes
3
answers
2k
views
Is every Schwartz function the product of two Schwartz functions?
A Schwartz function on $\mathbb R^d$ is a $C^\infty$ function, such that all differentials of order $k \ge 0$ decay faster than any polynomial. They include the class $C^\infty_c(\mathbb R^d)$ of ...
- 475
17
votes
2
answers
5k
views
Positive-Definite Functions and Fourier Transforms
Bochner's theorem states that a positive definite function is the Fourier transform of a finite Borel measure. As well, an easy converse of this is that a Fourier transform must be positive definite.
...
- 4,952
17
votes
2
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834
views
When is $\sum_{n\in\mathbb Z} f(x+n)$ constant?
A recently asked question (linked here) deals with the remarkable identity
$$ \sum_{n\in\mathbb Z} \mathrm{sinc}(n+x)= \pi,\quad x\in\mathbb R, $$
where $\mathrm{sinc}(x)=\sin(x)/x$.
It is easy ...
- 550
17
votes
4
answers
2k
views
What are the major differences between real and complex Banach space?
Most theorems under real Banach space settings have their twin brothers for complex ones, say, the Hahn-Banach theorem. However, some theorems are not valid in complex Banach spaces, and vice versa.
...
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