All Questions
13,926 questions
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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(\...
- 493
20
votes
2
answers
3k
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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?
- 28k
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?
- 1,638
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*-...
- 373
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2
answers
1k
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Rugged manifold
It is well known that any compact smooth $m$-manifold can be obtained from $m$-ball by gluing some points on the boundary.
Is it still true for topological manifold?
Comments:
To proof the smooth ...
- 45k
20
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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 ...
- 101
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3
answers
8k
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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 ...
- 757
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2
answers
1k
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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 ...
- 545
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1
answer
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A function composed with itself produces the identity
Let $B$ be the closed unit ball in $\mathbb R^3$ and $f: B\to B$ continuous, such that $f\circ f$ is the identity (i.e., $f\circ f=\mathbb 1_B$) and $f$ restricted on $\partial B$ is also the identity ...
- 2,933
20
votes
2
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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 ...
- 3,515
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3
answers
2k
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Stone–Čech Compactification of $\mathbb{Z}$ with Fürstenberg Topology
The Stone–Čech Compactification of $\mathbb{N}$ as a discrete space has been extensively studied and can be represented using ultrafilters.
Consider $X=(\mathbb{Z},\mathcal{T})$, where $\mathcal{T}$ ...
- 571
20
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2
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870
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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(\...
- 3,984
20
votes
2
answers
545
views
$\kappa$-homogeneous topological spaces
Let $\kappa>0$ be a cardinal and let $(X,\tau)$ be a topological space. We say that $X$ is $\kappa$-homogeneous if
$|X| \geq \kappa$, and
whenever $A,B\subseteq X$ are subsets with $|A|=|B|=\kappa$...
- 48.9k
20
votes
2
answers
1k
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An order type $\tau$ equal to its power $\tau^n, n>2$
(This is a re-post of my old unanswered question from Math.SE)
For purposes of this question, let's concern ourselves only with linear (but not necessarily well-founded) order types.
Recall that:
$...
- 6,819
20
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2
answers
2k
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Is every compact topological ring a profinite ring?
There are a lot of compact (Hausdorff) groups, whereas every compact field is finite. What about rings? Is there a classification theorem for compact rings? If you take a cofiltered limit of finite ...
- 2,210
20
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3
answers
2k
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Duality between topology and bornology
I want to understand in what sense topology is dual to bornology at a most basic level. Therefore, I rephrased the definition of a bornology in the following way:
Let $X$ be a set and let $\mathcal{P}(...
- 1,813
20
votes
1
answer
527
views
Combination topological space and locale?
The traditional theory of topological spaces (as formalized by Bourbaki) starts with a set of points, then builds a structure on that. In contrast, the theory of locales starts with a frame of opens (...
- 2,754
20
votes
4
answers
2k
views
Problems for developing mathematical visualization expertise
Einstein stated that he often explored and reasoned visually and spatially, and only after achieving understanding cast his insights into algebraic form. He could just "see" the answer. There are ...
20
votes
2
answers
2k
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Several questions about Gauss's mathematical conception of braids
I'm trying to figure out several things about Gauss's thoughts concerning a certain four-strand braid. The reference my questions are based on is mainly Moritz Epple's excellent article "orbits ...
- 2,099
20
votes
2
answers
691
views
A "dimension" for Tychonoff spaces
It's well-known that any Tychonoff space $X$ can be embedded in $[0,1]^k$ for some cardinal $k$. It's natural to ask what the smallest such $k$ is (let's call it $k(X)$). However, this probably ...
- 301
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3
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1k
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How thinly connected can a closed subset of Hilbert space be?
Let H be a separable (and infinite-dimensional) Hilbert space. Is it known whether there exists an infinite
subset C of H with the following properties.? (1) C is connected and closed in H. (2) No ...
- 6,215
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1
answer
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Topological embeddings of real projective space in euclidean space
I was wondering whether the real projective space $\Bbb{R}P^n$ embeds topologically into $\Bbb{R}^{n+1}$ for odd $n$.
It certainly doesn't for even $n$ because of Alexander duality. Also it doesn't ...
- 2,797
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2
answers
1k
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The Gelfand duality for pro-$C^*$-algebras
The Gelfand duality says that
$$X\to C(X)$$
is a contravariant equivalence between the category of compact Hausdorff spaces and continuous maps and the category of commutative unital $C^*$-algebras ...
- 1,344
20
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1
answer
993
views
Which spaces are characterized by functions with compact support ?
It's well known that two locally compact Hausdorff spaces $X, Y$ are homeomorphic iff the rings $C_0(X), C_0(Y)$ (continuous functions vanishing at infinity) are isomorphic.
Is there a class $\...
- 16.2k
20
votes
1
answer
2k
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Connected and locally connected, but not path-connected
Allow me to use some non-standard terminology:
A h-contractible space is a non-empty topological space $X$ such that, for any topological space $T$ and any pair of continuous maps $f_0, f_1 : T \to X$...
- 15.9k
20
votes
2
answers
922
views
A functional inequality about log-concave functions
Let $f,g$ be smooth even log-concave functions on $\mathbb{R}^{n}$, i.e.,$f=e^{-F(x)}, g=e^{-G(x)}$ for some even convex functions $F(x),G(x)$. Is it true that:
$$
\int_{\mathbb{R}^{n}} \langle \...
- 3,946
19
votes
4
answers
18k
views
On the series 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ...
It is well-known that
A: The series of the reciprocals of the primes diverges
My question is whether property A is in some sense a truth strongly tied to the nature of the prime numbers.
Property A ...
- 8,842
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
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3
answers
2k
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How many tacks fit in the plane?
Call a tack the one point union of three open intervals. Can you fit an uncountable number of them on the plane? Or is only a countable number?
- 193
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
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3
answers
2k
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Non-homeomorphic spaces such that taking away a point makes them homeomorphic
Are there topological spaces $X,Y$, each having more than $2$ points, satisfying the following two properties?
$X\not\cong Y$, and
there is a bijection $\varphi: X\to Y$ such that for all $x\in X$ ...
- 48.9k
19
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3
answers
5k
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Is a inverse limit of compact spaces again compact ?
Then one can construct a model for the inverse limit by taking all the compatible sequences.
This is a subspace of a product of compact spaces. This product is compact by Tychonoff. If all the spaces ...
- 11.1k
19
votes
4
answers
8k
views
Unique limits of sequences plus what implies Hausdorff?
It is known that there are non-Hausdorff spaces which admit unique limits for all convergent sequence (see here) and it is also known that unique limits for nets implies Hausdorff.
What I am ...
- 12.7k
19
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3
answers
1k
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"Anti" fixed point property
Let $(X,\tau)$ be a topological space. If $f:X\to X$ is continuous, we say $x\in X$ is a fixed point if $f(x) = x$.
The space $(X,\tau)$ is said to have the anti fixed point property (AFPP) if the ...
- 48.9k
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
4
answers
4k
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When is a finite cw-complex a compact topological manifold?
I think the statement of the question is pretty straightforward. Given a finite $n$-dimensional CW complex, are there necessary and sufficient conditions for determining that it is also a compact $n$-...
- 732
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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
2
answers
6k
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Is any function taking compact sets to compact sets, and connected sets to connected sets, necessarily continuous?
It is well-known that continuous image of any compact set is compact, and that continuous image of any connected set is connected.
How far is the converse of the above statements true?
More precisely:...
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2
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How bogus is the glitzy proof of Borsuk-Ulam?
Suppose $f: S^2 \rightarrow {\bf R}^2$ is continuous; let $A$ be the set of points $u \in S^2$ such that $f(u)-f(-u) \in {\bf R} \times \{0\}$ (where $-u$ denotes the antipode of $u$). Given $u,-u \in ...
- 19.7k
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2
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1k
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Are there space filling curves for the Hilbert cube?
There is a surjective continuous map $[0;1]\rightarrow [0;1]^2$ ("space filling curve"). Using such a map one can easily get space filling curves for all finite dimensional cubes.
So my question is: ...
- 11.1k
19
votes
3
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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
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votes
3
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Is there a Cantor set $C$ in $\mathbb{R}^{2}$ so the graph of every continuous function $[0,1]\rightarrow [0,1]$ intersects $C$?
Consider the Cantor ternary set on the real line with the usual topology and define a Cantor set to be any topological space $C$ homeomorphic to the Cantor ternary set.
The idea is to construct a ...
- 2,136
19
votes
4
answers
5k
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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
2
answers
2k
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Complete metric on the space of Jordan curves?
I was interested in putting a complete metric on the space of Jordan curves. Say, just planar Jordan curves contained in $B(\bar{0}, 2) \backslash B(\bar{0}, 1)$ which separates $\bar{0}$ and infinity....
- 375
19
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1
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2k
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Is the closed unit ball of the Hilbert space homeomorphic to the unit sphere ?
Is the closed unit ball of the Hilbert space (or, for that matter, of the Hilbert cube, in some metric) homeomorphic to the unit sphere (viz., its own boundary) ? This is clearly uncharacteristic of ...
- 1,445
19
votes
1
answer
556
views
Can an injective $f: \Bbb{R}^m \to \Bbb{R}^n$ have a closed graph for $m>n$?
Question. Suppose $m>n$ are positive integers. Is there a one-to-one $f: \Bbb{R}^m \to \Bbb{R}^n$ such that the graph $\Gamma_f$ of $f$ is closed in $\Bbb{R}^{m+n}$?
Remark 1. The answer to the ...
- 17.7k
19
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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
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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