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51 votes
5 answers
18k views

Integrability of derivatives

Is there a (preferably simple) example of a function $f:(a,b)\to \mathbb{R}$ which is everywhere differentiable, such that $f'$ is not Riemann integrable? I ask for pedagogical reasons. Results in ...
Mark Meckes's user avatar
  • 11.4k
44 votes
10 answers
47k views

Is square of Delta function defined somewhere?

I am wondering whether anyone knows if the square of Dirac Delta function is defined somewhere. In the beginning, this question might look strange. But by restricting the space of the test functions, ...
44 votes
3 answers
4k views

Smooth functions for which $f(x)$ is rational if and only if $x$ is rational

A friend of mine introduced me to the following question: Does there exist a smooth function $f: \mathbb{R} \to \mathbb{R}$, ($f \in C^\infty$), such that $f$ maps rationals to rationals and ...
J. J.'s user avatar
  • 543
38 votes
26 answers
57k views

Text for an introductory Real Analysis course.

Any suggestions on a good text to use for teaching an introductory Real Analysis course? Specifically what have you found to be useful about the approach taken in specific texts?
33 votes
3 answers
3k views

Reference request for translating from Top to C*-alg

Some recent questions on MO (for example, Do subalgebras of C(X) admit a description in terms of the compact Hausdorff space X?) have been about Gelfand duality — namely, that the categories of ...
Matthew Daws's user avatar
  • 18.7k
33 votes
2 answers
2k views

What is the smallest set of real continuous functions generating all rational numbers by iteration?

I recently came across this problem from USAMO 2005: "A calculator is broken so that the only keys that still work are the $\sin$, $\cos$, $\tan$, $\arcsin$, $\arccos$ and $\arctan$ buttons. The ...
Ivan Meir's user avatar
  • 4,862
30 votes
4 answers
2k views

is f a polynomial provided that it is "partially" smooth?

Let $f$ be a $C^\infty$ function on $(c,d)$ ,and let $O=\cup_{n\in \mathbb{Z}^+} (a_n,b_n)$ where $(a_n,b_n)$ are disjoint open interval in $(c,d)$ and $O$ is dense in $(c,d)$. Suppose for each $n\in ...
Ben's user avatar
  • 407
29 votes
1 answer
1k views

About the function $\prod_{k \in \mathbb{N}}(1-\frac{x^3}{k^3})$

I'm wondering if the function $$f(x)=\prod_{k \in \mathbb{N}}\left(1-\frac{x^3}{k^3}\right)$$ has a name, or if there are any properties (especially about derivatives of $f$) have studied so far. I ...
droptable's user avatar
  • 483
28 votes
4 answers
3k views

"Converse" of Taylor's theorem

Let $f:(a,b)\to\mathbb{R}$. We are given $(k+1)$ continuous functions $a_0,a_1,\ldots,a_k:(a,b)\to\mathbb{R}$ such that for every $c\in(a,b)$ we can write $f(c+t)=\sum_{i=0}^k a_i(c)t^i+o(t^k)$ (for ...
Mizar's user avatar
  • 3,146
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 ...
Andreas Rüdinger's user avatar
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 ...
NPC's user avatar
  • 309
23 votes
4 answers
5k views

Are proper linear subspaces of Banach spaces always meager?

Let X be a Banach space, and let Y be a proper non-meager linear subspace of X. If Y is not dense in X, then it is easy to see that the closure of Y has empty interior, contradicting Y being non-...
Brandon Seward's user avatar
23 votes
1 answer
2k views

Which Fréchet spaces have a dual that is a Fréchet space?

I've read the claim that Fréchet spaces that are not Banach spaces never have a dual that is a Fréchet space, but have not been able to find a proof of this statement. Is it trivial or does someone ...
Tim van Beek's user avatar
  • 1,544
23 votes
5 answers
6k views

Hahn-Banach without Choice

The standard proof of the Hahn-Banach theorem makes use of Zorn's lemma. I hear that, however, Hahn-Banach is strictly weaker than Choice. A quick search leads to many sources stating that Hahn-Banach ...
Mark Kim-Mulgrew's user avatar
23 votes
3 answers
1k views

Which $\ast$-algebras are $C^\ast$-algebras?

It's well-known that the norm on a $C^\ast$-algebra is uniquely determined by the underlying $\ast$-algebra by the spectral radius formula. Therefore there should be a way to axiomatize $C^\ast$-...
Tim Campion's user avatar
  • 63.9k
20 votes
6 answers
7k views

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 ...
Mikhail Katz's user avatar
  • 16.6k
20 votes
2 answers
4k views

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 ...
user18107's user avatar
  • 101
20 votes
3 answers
1k views

mixing convex and concave for convexity

Let $n\in\mathbb{N}$ and $0<x<1$ be a real number. Is the following a convex function of $x$? $$G_n(x)=\log\left(\frac{(1+x^{4n+1})(1+x^{4n-1})(1+x^{2n})(1-x^{2n+1})}{(1+x^{2n+1})(1-x^{2n+2})}\...
T. Amdeberhan's user avatar
19 votes
2 answers
2k views

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 ...
Ivan Feshchenko's user avatar
18 votes
1 answer
2k views

Function of two sets intersection

Let $U$ be the set of all nonempty subsets of $[0,1]$ that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Does ...
pi66's user avatar
  • 1,209
16 votes
1 answer
2k views

What (classes of) Banach spaces are known to have Schauder basis?

Motivation: I am trying to see for what class of Banach spaces the following result is true: There exists an increasing sequence of finite dimensional subspace {$V_n$} of a Banach space X (with ...
Clark Chong's user avatar
16 votes
4 answers
2k views

Is there a maximum to the amount of disjoint non-measurable subsets of the unit interval with full outer measure?

This question arose a few years back when I was an assistant teacher on a course of basic (Lebesgue) measure theory, but I didn't find an answer or anyone able to solve the problem. The setting of the ...
Rami Luisto's user avatar
16 votes
5 answers
3k views

Measure theory treatment geared toward the Riesz representation theorem

I'm looking for recommendations for books (or lecture notes) that develop measure theory in sufficient detail to state and prove the Riesz representation theorem (which is the characterization of the ...
Igor Khavkine's user avatar
15 votes
4 answers
3k views

Universal $C^*$-algebra with generators and relations

We say that the $C^*$-algebra $A$ generated by $a_1,...,a_n$ is universal subject to relations $R_1,...,R_m$ if for every $C^*$-algebra $B$ with elements $b_1,...,b_n$ satisfying relations $R_1,...,...
truebaran's user avatar
  • 9,330
14 votes
1 answer
2k views

Infinite tensor product of states

Tensor products of finite number of different objects are always well described in the literature. However, the situation of infinite tensor products seems to be much tougher. Even in the simplest ...
Glacier's user avatar
  • 143
13 votes
5 answers
3k views

Reference request: Oldest calculus, real analysis books with exercises?

Per the title, what are some of the oldest calculus, real analysis books out there with exercises? Maybe there are some hidden gems from before the 20th century out there. Edit. Unsolved exercises ...
13 votes
2 answers
1k views

Calkin Algebra and the embedding

Let $H$ be a separable, infinite dimensional Hilbert Space and $Calk(H):=B(H)/K(H)$ denotes the Calkin algebra. There is obvious surjection $\pi: B(H) \to Calk(H)$ but I'm interested in somehow ...
truebaran's user avatar
  • 9,330
12 votes
2 answers
2k views

Function and Fourier transform vanish on an interval

I'm no expert on these things (and this may not be cutting edge research level; it's really motivated by this MSE question), but it seems that there are non-zero measures (and also functions (?), I ...
Christian Remling's user avatar
12 votes
2 answers
3k views

Direct proof of injectivity of $L_\infty$

I would like to know a simple proof of isometric injectivity of $L_\infty$. The proof I've found in Topics in Banach space theory. F. Albiac, N. Kalton uses two deep result. $L_\infty$ as ...
Norbert's user avatar
  • 1,697
11 votes
4 answers
1k views

Norm continuous infinite dimenisonal representation of a Lie group

Given a Lie group G and an infinite dimensional Hilbert space $\mathcal{H}$. In the literature I have only encountered the two following notions of a representation $\pi$ of G on $\mathcal{H}$ : 1) $\...
jsb's user avatar
  • 403
11 votes
1 answer
1k views

Stone-Weierstrass analogue for $L^p$

Let $A$ be a complex algebra of bounded measurable functions on the measure space $(X,\mu)$ (case of $[0,1]$ with Lebesgue measure is enough for me) closed under conjugation. Assume that $A$ separates ...
Fedor Petrov's user avatar
10 votes
2 answers
826 views

Is there a measure on $[0,1]$ that is 0 on meagre sets and 1 on co-meagre sets

I'm curious if there is a finite measure on the $\sigma$-algebra of subsets of $[0,1]$ with the Property of Baire, whose null sets are exactly the meagre sets. I'd also be interested how "nice" such ...
Tom Bouley's user avatar
10 votes
2 answers
1k views

Proof in constructive mathematics that the principal square root function exists in any Cauchy complete Archimedean ordered field

In classical mathematics, there exists only one Cauchy complete Archimedean ordered field, the Dedekind complete Archimedean ordered field. However, in constructive mathematics, there are multiple ...
Madeleine Birchfield's user avatar
10 votes
3 answers
739 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
9 votes
3 answers
2k views

Generalizations and relative applications of Fekete's subadditive lemma

Fekete's (subadditive) lemma takes its name from a 1923 paper by the Hungarian mathematician Michael Fekete [1]. A historical overview and references to (a couple of) generalizations and applications ...
Salvo Tringali's user avatar
9 votes
1 answer
2k views

Density of smooth functions on Hölder spaces

The following result is often cited without reference in the context of PDEs: Let $\varOmega \subset\mathbb R^n$ be a bounded open set with smooth boundary. If $0<\beta<\alpha<1$ then $C^\...
Nautilus's user avatar
  • 727
8 votes
1 answer
380 views

Lavrentiev phenomenon between $C^1$ and Lipschitz

Does there exist a (onedimensional) integral functional of calculus of variations (with $f$ finite everywhere) $$ F(y)=\int_a^b f(t,y(t),y'(t))\,dt
 $$ such that $$ \inf_{y\in Lip([a,b])}F(y)<\inf_{...
Carlo Mantegazza's user avatar
8 votes
1 answer
716 views

A non-hyperfinite type III factor from an action of the free group on the circle

We define below a von Neumann algebra $\mathcal{M}$ from an action of the free group on the circle, and we prove that $\mathcal{M}$ is a non-hyperfinite type ${\rm III}$ factor. Question : Is $\...
Sebastien Palcoux's user avatar
8 votes
4 answers
4k views

Non-zero smooth functions vanishing on a Cantor set

It is easy to give examples of continuous functions $f:[0,1]\to \mathbb R_+\cup\{0\}$ non-zero but vanishing on a Cantor set (ex: Can Cantor set be the zero set of a continuous function?). It is ...
user39115's user avatar
  • 1,805
7 votes
3 answers
841 views

Distance function to $\Omega\subset\mathbb{R}^n$ differentiable at $y\notin\Omega$ implies $\exists$ unique closest point

I am trying to show the following two statements are true: (1) For any nonempty set $\Omega\subset\mathbb{R}^n$, the set $B$ consisting of points $y\notin\Omega$ where there is not a unique closest ...
L P's user avatar
  • 323
7 votes
2 answers
582 views

"sinc-ing" integral

Let $a_1,\dots,a_n, b$ be positive real numbers. *Question.** Is this true? $$\int_{-\infty}^{\infty}\frac{\sin(bx+a_1x+\cdots+a_nx)}{x}\prod_{j=1}^n\frac{\sin(a_jx)}{a_jx}\,\,dx=\pi.$$ My ...
T. Amdeberhan's user avatar
7 votes
1 answer
1k views

Properties of convolutions

Consider the function $$f_{n}(x)=e^{-x^2}x^n.$$ and the function $$h_p(x):=e^{-\vert x \vert^p}.$$ My goal is to analyze $$ F_p(y):=\frac{(f_2*h_p)(y)}{(f_0*h_p)(y)}- \left(\frac{(f_1*h_p)(y) }{(f_0*...
Landauer's user avatar
  • 173
6 votes
1 answer
474 views

Proof that $L^2(0,T;X)^* = L^2(0,T;X^*)$

How is the proof that $$[L^2(0,T;X)]' = L^2(0,T;X')$$ looking like, where $X$ is a Hilbert space? I am asking for the proof that the dual space of $L^2(0,T;X)$ is the space $L^2(0,T;X^*)$. Is the ...
lollypop's user avatar
6 votes
1 answer
2k views

Comparing norms on tensor products of matrices

Given a Hilbert space $H$, let $S_1(H)$ denote the space of trace-class operators on $H$, with the trace-class norm or Schatten 1-norm. That is $$ \Vert T \Vert_1 = \sum_{j\geq 1} |s_j| $$ where $(s_1,...
R.N's user avatar
  • 209
6 votes
1 answer
1k views

Symmetric basis of harmonic homogeneous polynomials

Recently, a question about the beautiful theory of harmonic polynomials made me aware there is something I've wanted to know for a long time. As is well known, for any number of variables $n$ and any ...
Pietro Majer's user avatar
  • 60.5k
6 votes
1 answer
696 views

Reference request: optimal $L^p$ regularity for solutions to $-\Delta u=f$ with $f\in L^1(R^d)$

The tilte says it all. Given $f\in L^1(R^d)$ (let me restrict to dimension $d\geq 3$ for convenience), what is the optimal $L^p$ regularity for solutions to $$ -\Delta u=f\hspace{3cm}(1)? $$ I'm of ...
leo monsaingeon's user avatar
5 votes
0 answers
198 views

Heuristic and graphic representation of BV functions and their singularities

This question is about some heuristics and graphs of BV functions. In 1-dimensional setting, two key examples of $BV$ functions $u: \mathbb R \to \mathbb R$ are the Heaviside function, whose ...
Riku's user avatar
  • 839
5 votes
2 answers
242 views

Can we stay invertible while approximating linear maps in Sobolev spaces?

Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain with a smooth boundary. Fix $1<p<n$. Let $A \in W^{1,p}(\Omega;\text{End}(\mathbb{R}^n)) \cap C(\Omega;\text{End}(\mathbb{R}^n))$ ...
Asaf Shachar's user avatar
  • 6,741
5 votes
2 answers
1k views

Is the space of signed finite measures on a compact set $M([0,1])$ a sequential space?

Let $M([0,1])$ be the set of finite signed measures on $[0,1]$ (with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all $\...
Ori's user avatar
  • 95
5 votes
1 answer
395 views

Universal decay rate of the Fisher information along the heat flow

I'm looking for a reference for the following fact: In the torus $\mathbb T^d$ let me denote by $u_t=u(t,x)$ the (unique, distributional) solution of the heat equation $$ \partial_t u=\Delta u $$ ...
leo monsaingeon's user avatar

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