All Questions
10,934 questions
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Do subalgebras of C(X) admit a description in terms of the compact Hausdorff space X?
In light of the well-known theorem of Gelfand that, bluntly put, ends up saying that unital abelian C*-algebras are the 'same' as compact Hausdorff topological spaces, I tried to compile a dictionary ...
- 1,105
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4
answers
2k
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Why the unreasonable applicability of complex numbers in physics/engineering? [duplicate]
After years of using complex numbers in every kind of analysis of physical and electrical engineering problems I am starting to wonder: why is this particular algebra so effective in modelling the ...
- 217
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2
answers
490
views
Surjective linear isometries on $\ell_\infty(\mathbb{N})$
In Volume 1 of "Classical Banach Spaces" Lindenstrauss and Tzafriri note that all surjective linear isometries on $\ell_\infty$ are of the from $(a_i) \mapsto (\varepsilon_i a_{\pi(i)})$ ...
- 1,073
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3
answers
739
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Is there a version of Fischer-Riesz theorem for Banach space?
$( \Omega,F, P )$: a measurable space equipped with a finite measure
$(B , \Vert \cdot \Vert) $ : a Banach space with $\mathcal{B}$ as its borelian $\sigma$-algebra
$p$ : a constant bigger than $1$
...
- 406
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5
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5k
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Applications of functional analysis beyond analysis(towards algebra, geometry, number theory...) [closed]
So far, We have seen the applications of functional analysis in PDE, probability and many areas in applied mathematics. On the other hand, methods of algebraic topology are introduced to functional ...
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2
answers
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Why does Riesz's Representation Theorem apply in quantum mechanics?
$\DeclareMathOperator\tr{tr}$One begins with a quantum mechanical system, i.e. a unital $C^*$-algebra $A$.
It is common to begin the discussion with embedding $A$ into the algebra of bounded operators ...
- 2,071
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2
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1k
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Fourier series of $\log(a +b\cos(x))$?
By numerical computation it seems like, if $a_0 < a_1$:
$$
\begin{multline}
\log({a_0}^2 + {a_1}^2 + 2 a_0 a_1 \cos(\omega t)) = \log({a_0}^2 + {a_1}^2) \\
+ \frac{a_0}{a_1}\cos(\omega t)
- \frac{...
10
votes
1
answer
833
views
This is not a dyadic cosine-product
The double-angle formula, $\sin2x=2\sin x\cos x$, turns the scary-looking integral
$$\int_0^{\infty}dz\prod_{k=1}^{\infty}\cos\frac{z}{2^k}$$
into fun once you realize $\prod_k\cos\frac{z}{2^k}=\frac{\...
- 43.2k
10
votes
3
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671
views
Is there a continuous analogue of Ramanujan graphs?
I think it might help to think of the following definition of a Ramanujan graph - a graph whose non-trivial eigenvalues are such that their magnitude is bounded above by the spectral radius of its ...
- 617
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4
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2k
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Reading for finite Fourier analysis
Can anyone recommend some good reading for Fourier analysis (and the Fourier transform) over finite abelian groups? I've found it given brief descriptions in both books on representation theory and on ...
- 7,013
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2
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804
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General recipe for building C*-algebras out of combinatorial object
I want to ask what should be a nice way to build C*-algebras out of objects like groups, inverse-semigroups, semigroups, ringgs or graphs. I know there are well known construction of C*-algebras out ...
- 493
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2
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1k
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On equibounded sequences in $L^\infty$
Let $f_n: [0, 1] \to \mathbb R$ be a sequence of positive functions in $L^\infty$ (hence a fortiori in $L^1$) that are equibounded in $L^\infty$ norm - that is $\sup_{n \in \mathbb N} \|f_n\|_{L_\...
- 6,213
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1
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509
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A quantity measuring the separability of Banach spaces
Let $X$ be a Banach space. It is natural for us to introduce a quantity measuring the separability of sets as follows: for a subset $A$ of $X$, we set
$\textrm{sep}(A)=\inf\{\epsilon>0: A\subseteq ...
- 3,343
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2
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3k
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Absolute continuity on $R^{n}$
I know the definition of absolute continuity if there is a function $f:(a,b)\rightarrow R$.
I wonder what is an analogy of this concept if we have a function $f:A\rightarrow R$, where $A\subset R^{n}$ ...
- 399
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1k
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Proving Poincaré's inequality for Boolean functions over the hypercube without Fourier analysis
$\DeclareMathOperator\Inf{Inf}\DeclareMathOperator\unif{unif}$I have been attempting to find a non-Fourier-analytic proof of Poincaré's inequality for Boolean functions over the hypercube. Let's call ...
- 261
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2
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1k
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Cone shaped solutions to wave equation
When I studied physics, we learned how to write down planar waves and spherical waves. But, when I turn on my flashlight, I see a cone of light. How can I see that there is a solution to the wave ...
- 156k
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1
answer
2k
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Quantum functional analysis
Can one explain some philosophy behind "quantum functional analysis" (or "quantized functional analysis") which was initiated and developed by such researchers as: Ruan Z.-J., Pisier J., Effros E.G., ...
- 537
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3
answers
1k
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References: spectral analysis of the Laplacian operator
I'm looking for several references on the spectral analysis of the Laplacian operator. It is such a well-known topic, but I'm a bit struggling to locate modern systematic expositions in the literature....
user89890
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1
answer
658
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Are functions of moderate growth a bornological space?
I was thinking a bit about distribution theory the last weeks and stumbled across the following question:
There are two natural locally convex topologies on the space of smooth functions of moderate ...
- 9,650
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2
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3k
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Cesaro means and Banach limits
Consider the class of bounded sequences to which every Banach limit (non-negative shift-invariant continuous functional on $l^\infty$ taking convergent sequences in the usual sense to their limits) ...
- 217
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2
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1k
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What function is this? -Counterexample found: it is not Lipschitz-
THE FRAMEWORK
Let $0<\lambda\le1$ and consider
$$
\Psi:(\Bbb R[X]_0,||\cdot||_{\lambda})\longrightarrow(\mathcal C^{\lambda}[0,1],||\cdot||_{\lambda})
$$
defined as
$$
\Psi(p):=\sup_{0\le u\le\...
- 779
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2
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5k
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Direct proof of the separation theorem of Hahn-Banach
The "extension" (or "analytic") form of the theorem of Hahn-Banach has a natural and yet elegant proof. In just any textbook I have ever seen, it is proved first; the "separation" (or "geometric") ...
- 3,388
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957
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Do eigenfunctions of elliptic operator form basis of $H^k(M)$?
We know that the eigenfunctions of the Laplacian on a compact manifold $M$ form a countable basis of $H^1(M)$ and $L^2(M)$.
If $L$ is a $2k$-order elliptic operator, do the eigenfunctions of $L$ ...
- 103
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3
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1k
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Compact subgroups of the unitary group of operators in a hilbert space
Is there a characterization for the compact subgroups of the unitary operators in a Hilbert space, where the unitaries are furnished with the norm topology? What about other topologies?
- 101
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559
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Can Birkhoff's ergodic theorem for integrable functions easily be deduced from Birkhoff's ergodic theorem for bounded functions?
It seems to me that a considerably simpler proof [see below] of Birkhoff's ergodic theorem can be obtained for bounded observables than for more general $L^1$ observables. Therefore, I feel like it ...
- 708
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973
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$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$
I posted this question first in Math.StackExchange one week ago here, but I didn't get an answer or a helpful comment so I repost it here:
Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded ...
- 261
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594
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Existence of a strongly continuous topologically irreducible representation of a compact group on an infinite dimensional Banach space?
Does there exists a triple $(G, X, \pi)$, where $G$ is a compact group, $X$ an infinite dimensional Banach space over $\mathbf{C}$, and $\pi : G \to B(X)$ a strongly continuous representation of $G$, ...
- 960
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739
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Unconditionally convergent series in some functional spaces
Linked with this question and discussion
(Bilinear product of two summable families), I am very
interested in counterexamples/results about the following questions (cf the end).
First, I recall that a
...
- 3,738
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2
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926
views
Continuity of the product map
Let $A$ be a $C^*$-algebra.
Is it possible to characterize $A$ for which the product map defined by
$$\sum\limits_{i=1}^n a_i\otimes b_i \mapsto \sum\limits_{i=1}^n a_i b_i$$
is continuous with ...
- 4,705
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881
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volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?
We denote by $\otimes_{\epsilon}$ the injective Banach tensor product.
Which is the asymptotic volume of the unit ball of the Banach space $\ell_1^n\otimes_{\epsilon}\ell_1^n$?
- 1,222
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666
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Reference request: Extensions of Wiener's Tauberian Theorem
Wiener's Tauberian Theorem says that linear combinations of translations of a function $f$ are dense in $L^1(\mathbb{R})$ if and only if the zero set of the Fourier transform of $f$ is empty. This is ...
- 710
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606
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A characterization of metric spaces admitting a bi-Lipschitz embedding into a Hilbert space?
Theorem (??) derived in this MO-post from Schoenberg's theorem yeilds a "bipartite" characterization of metric spaces that admit an isometric embedding into a Hilbert space. This Theorem (??)...
- 41.9k
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2
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1k
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Harmonic oscillator discrete spectrum
Let us act intentionally stupid and assume we do not know that we can solve for the spectrum of the harmonic oscillator
$$-\frac{d^2}{dx^2}+x^2$$
explicitly.
Is there an abstract argument why the ...
- 501
10
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2
answers
1k
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Bounding exponential sum with square roots
It is well known that for each $m\in\mathbb{N}$
$$\lim_{N\to\infty}\frac1N\sum_{n=1}^Ne^{2\pi i\sqrt{nm}}=0$$
My question is whether there is some uniformity in the variable $m$.
More precisely, is it ...
- 1,701
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6
answers
6k
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Fourier transform of (real) exponential
Is it possible to make sense, in distributional sense, of the Fourier transform of the exponential function (defined over the whole real line)?
- 101
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3
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861
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Takesaki theorem 2.6
I originally posted this question on MSE and didn't get a satisfactory answer, even after putting a bounty on it. Hence, I thought I should ask here:
Consider the following theorem in Takesaki's book &...
- 175
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7
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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
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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
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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
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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
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368
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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
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574
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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
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594
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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
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915
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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
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809
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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
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929
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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
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652
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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
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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
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1
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521
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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
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1
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694
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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