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Extracting a common convergent indexing from an uncountable family of sequences

Let $\mathcal{A}$ be some uncountable index set and $X$ be some separable reflexive Banach space. For each $\alpha \in \mathcal{A}$, let \begin{equation} \{ x_n^{\alpha} \}_{n=1}^\infty \end{equation} ...
Isaac's user avatar
  • 3,477
10 votes
3 answers
2k views

Pathological product space norm

Let $X$ and $Y$ be two normed vector spaces and $n(\cdot, \cdot)$ be any norm on $\mathbb{R}^2$. Is it always possible to define a norm on the product vector space $X \times Y$ as $||(x, y)||_{X \...
Zuza's user avatar
  • 202
10 votes
3 answers
834 views

Rigorous justification that overdetermined systems do not have a solution

There is the following well known and very useful heuristic principle: Assume one has a natural map from the space of $k$-tuples of functions in $n$ variables into the space of $K$-tuples of functions ...
asv's user avatar
  • 21.8k
10 votes
5 answers
2k views

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 ...
Joshua Seaton's user avatar
10 votes
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 $(a_i) \mapsto (\varepsilon_i a_{\pi(i)})$ ...
Kevin Beanland's user avatar
10 votes
3 answers
741 views

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 $1$ ...
Taro Tokyo's user avatar
10 votes
5 answers
5k views

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 ...
10 votes
2 answers
2k views

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 ...
Andrew NC's user avatar
  • 2,071
10 votes
3 answers
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 ...
Student's user avatar
  • 617
10 votes
2 answers
804 views

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 ...
SiOn's user avatar
  • 493
10 votes
2 answers
1k views

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_\...
Nate River's user avatar
  • 6,285
10 votes
1 answer
509 views

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 ...
Dongyang Chen's user avatar
10 votes
2 answers
3k views

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 $A\subset R^{n}$ ...
Nikita Evseev's user avatar
10 votes
1 answer
2k views

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., ...
Fedor Goncharov's user avatar
10 votes
3 answers
1k views

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....
user avatar
10 votes
1 answer
659 views

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 ...
Johannes Hahn's user avatar
10 votes
2 answers
3k views

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) ...
kap44's user avatar
  • 217
10 votes
2 answers
1k views

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\...
Joe's user avatar
  • 779
10 votes
2 answers
5k views

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") ...
Delio Mugnolo's user avatar
10 votes
1 answer
957 views

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 $L$ ...
michael faber's user avatar
10 votes
3 answers
1k views

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?
Nicolas Börger's user avatar
10 votes
2 answers
559 views

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 ...
Julian Newman's user avatar
10 votes
1 answer
974 views

$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ 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 ...
Svetoslav's user avatar
  • 261
10 votes
2 answers
594 views

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 $G$, ...
Hua Wang's user avatar
  • 960
10 votes
2 answers
740 views

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 ...
Duchamp Gérard H. E.'s user avatar
10 votes
2 answers
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 ...
Kate Juschenko's user avatar
10 votes
2 answers
881 views

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 $\ell_1^n\otimes_{\epsilon}\ell_1^n$?
BigBill's user avatar
  • 1,222
10 votes
2 answers
669 views

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 ...
JohnA's user avatar
  • 710
10 votes
2 answers
606 views

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 (??)...
Taras Banakh's user avatar
10 votes
2 answers
1k views

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 ...
Zinkin's user avatar
  • 501
10 votes
3 answers
861 views

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 &...
Andromeda's user avatar
  • 175
10 votes
7 answers
1k views

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 ...
stefanarno's user avatar
10 votes
1 answer
1k views

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 ...
Matthew Daws's user avatar
  • 18.7k
10 votes
5 answers
4k views

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 ...
Matthew Daws's user avatar
  • 18.7k
10 votes
5 answers
1k views

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 ...
AgCl's user avatar
  • 2,745
10 votes
1 answer
368 views

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 ...
Matthias Ludewig's user avatar
10 votes
1 answer
576 views

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 ...
Jandré Snyman's user avatar
10 votes
1 answer
594 views

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 ...
Vesselin Dimitrov's user avatar
10 votes
1 answer
915 views

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 ...
Walt van Amstel's user avatar
10 votes
1 answer
930 views

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 ...
Dany Galicer's user avatar
10 votes
1 answer
652 views

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 ...
Nate Eldredge's user avatar
10 votes
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\...
Bas Spitters's user avatar
10 votes
1 answer
522 views

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 ...
Gabriel Palau's user avatar
10 votes
1 answer
695 views

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 ...
Estwald's user avatar
  • 1,391
10 votes
2 answers
504 views

Generalizations of the Robbins lemma and Gaussian integration by parts

This is getting no attention, so I'll try this here: The Robbins lemma, named after Herbert Robbins, says that if $X\sim\operatorname{Poisson}(\lambda)$ and $g$ is a function for which $\operatorname{...
Michael Hardy's user avatar
10 votes
3 answers
1k views

Historical developement of analysis and partial differential equations (especially in the 20th century)

Q: Is there a set of some comprehensive surveys or monographs describing (in technical detail) the historical development of the various subareas of analysis and partial differential equations? I'...
10 votes
1 answer
442 views

A question about tiling Hilbert Space

Let H be an infinite dimensional and separable Hilbert Space. Let e be a positive real number-which can be arbitrarily small. Does there exist a denumerably infinite set S of pairwise disjoint and ...
Garabed Gulbenkian's user avatar
10 votes
1 answer
1k views

Dual space of continuous Banach-space-valued functions

Let $X$ be a Banach space and $K$ some compact Hausdorff space. I am interested in the dual space of the Banach space $$C(K; X) = \lbrace f: K \to X, \ f \text{ is continuous}\rbrace, \qquad \lVert ...
Yaddle's user avatar
  • 381
10 votes
2 answers
281 views

Weak* continuity of positive parts

I'm a little embarrassed to be asking this, but surely there is a simple argument that I didn't see? Let $(f_\lambda)$ be a net in $l^\infty$ which converges weak* to $f \in l^\infty$. We do not ...
Nik Weaver's user avatar
  • 42.8k
10 votes
2 answers
1k views

The intuition behind the Hilbert projective metric and the Perron Frobenius Theorem

Recently I have read a proof of the Perron Frobenius Theorem for positive aperiodic matrices. In this proof, the trick is to put a metric in the "positive quadrant" of $\mathbb{R}^n$, $\mathbb{R}^{n}_+...
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