All Questions
9,958 questions
2
votes
3
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views
Show a linear operator is not compact
For $f\in L^2(0,\infty),$ define $(Tf)(x)=x^{-1}\int_0^x f(s)ds,$ for $x\in(0,\infty),$ then from hardy's inequality, $T\in B(L^2),$ my question is how to show that $T$ is not compact?
- 4,060
8
votes
1
answer
381
views
Estimating flat norm distance from a planar disc
Let $D\subset\mathbb R^2\subset\mathbb R^n$ be a unit planar disc in $\mathbb R^n$. Let $S$ be an orientable two-dimensional surface in $\mathbb R^n$ such that $\partial S=\partial D$. Of course, we ...
- 45k
2
votes
2
answers
317
views
Bibliography for topologies defined by a family of seminorms
Hello
I am trying to learn more about Fréchet spaces (in order to study the theory of distributions) and was wondering what people thought was the best resource.
Thank you very much.
- 27.7k
5
votes
1
answer
1k
views
Orthogonal complements in Hilbert bundles
It's a standard fact that for a finite-dimensional vector bundle with an inner product, the othogonal complement of any subbundle is itself a locally trivial vector bundle.
What is known about the ...
- 26.8k
6
votes
1
answer
1k
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Reference needed for: every idempotent in a C*-algebra is similar to a hermitian one
The result stated in the title is thoroughly standard - or that's the impression I got.
I seem to remember seeing it stated somewhere in a book I was reading in the library, and then reverse-...
- 7,329
0
votes
1
answer
288
views
The Quantum Operations On The Bipartite Systems
Given two distinct and noninteracting quantum mechanical
systems $\mathfrak{S}\_1$ and $\mathfrak{S}\_2$ with state spaces
$\mathcal H\_1$ and $\mathcal H\_2$, respectively, the state space
of the ...
- 6,342
3
votes
3
answers
2k
views
Error analysis of implicit functions
I'm trying to do propagation of error using the linearized variance method (assuming independent variables, thus no need for the covariance terms):
$$\sigma^2_f = \sum^n_{k=0} \left(\frac{\partial f}{...
- 15.4k
5
votes
1
answer
467
views
Info about Elton–Odell theorem
Hello everyone, could anyone please tell me where can I find information about the Elton–Odell theorem?
It states:
For any infinite dimensional Banach space $X$ there is a $q > 1$ so that $X$ ...
- 9,366
4
votes
1
answer
822
views
What is the tensor product of $L^p(\bf R)$ with $L^q(\bf R)$?
I'm wondering: What is the tensor product of $L^p({\bf R})$ with $L^q({\bf R})$?
(For p=q=2, the answer clearly should be $L^2({\bf R}^2)$; for other values of $p$ and $q$, it is not at all obvious ...
- 11.4k
3
votes
3
answers
584
views
Polynomials and L^p(R)
As someone who mostly does symbolic computation, I've always been puzzled by the fascination mathematicians seem to have with Lp(R) (for p<∞)? To be more precise, there are no non-trivial ...
- 2,283
3
votes
1
answer
1k
views
characterization of continuous functionals in weak-star topology
Reading Wojtaszczyk's Banach spaces for analysts, I'm trying to understand his proof that the space of all continuous linear functionals on $(X^\star,\sigma(X^\star, X))$ is $X$.
To show the $ \...
- 31.5k
4
votes
2
answers
340
views
Embeddings of Weighted Banach Spaces
Let be $d$ a positive integer, $\Omega=\mathbb{R}^{\mathbb{Z}^d}$ and fix $R\geq 2$. We define weighted Banach spaces
$$ \Omega_p:=\left\{ x\in \Omega\left| \left[\sum_{i\in\mathbb{Z}^d}\frac{|x_i|^...
- 25.8k
5
votes
1
answer
403
views
Nonlinear Nuclear Operators ?
Is there a "right" definition of the nuclear
operator in the nonlinear framework ? Of course, such an operator
must be compact, while a linear operator should be "nonlinearly"
nuclear iff it is ...
- 31.5k
24
votes
1
answer
2k
views
How many ways are there to globalize Harish Chandra modules?
Suppose $G$ a reductive Lie group with finitely many connected components, and suppose in addition that the connected component $G^0$ of the identity can be expressed as a finite cover of a linear Lie ...
CommunityBot
- 1
8
votes
3
answers
606
views
Compact Hausdorff and C^*-algebra "objects" in a category.
This is yet more on "algebraic objects in functional analysis".
Since Compact Hausdorff spaces are algebraic over Set, it seems to follow that one can find "Compact Hausdorff objects" in any suitable ...
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16
votes
0
answers
1k
views
Finite Rank Commutators
My former student Detelin Dosev and I are interested in classifying the commutators in $L(X)$, the bounded linear operators on the Banach space $X$ (see our joint paper on my home page or the ArXiv ...
- 31.5k
5
votes
3
answers
1k
views
Functional calculus for direct integrals
Suppose I have a direct integral of Hilbert spaces $H = \int^\oplus H_x dx $, and suppose I have an operator $T: H \to H$ which is decomposable, and so it can be written as
$T = \int^\oplus T_x$ for ...
- 1,540
9
votes
2
answers
1k
views
Borsuk pairs of Banach spaces
Given $X$, $Y$ two real Banach spaces, let's say that $(X,\ Y)$
is a Borsuk pair if for any continuous mapping $T$ : {$x$ $\in$
$X$ ; $||x||\leq1$} $\rightarrow$ $Y$ s.t. $T$ is odd on {$x$
$\in$ $X$ ;...
CommunityBot
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7
votes
2
answers
684
views
Yet more on distortion
I would like to elaborate a little bit on my previous question which can be found
here.
Firstly, let me recall that a separable Banach space $(X, \| \cdot \|)$ is said to be
arbitrarily distortable ...
CommunityBot
- 1
18
votes
3
answers
2k
views
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 ...
CommunityBot
- 1
9
votes
1
answer
996
views
Topological "Interpolation" ?
Let E be a normed space, and let $T$:E * $\rightarrow$ E * be
a nonlinear operator.
Suppose that :
1) $T$ is continuous from (E *, ||.||) to itself (i.e., it is norm-continuous).
and
2) $T$ is ...
- 25.8k
3
votes
2
answers
416
views
Which Banach spaces have categorical duals?
I was looking carefully at all the definitions, trying to understand exactly what was going on in this question on categorical duals in Banach spaces. It seems that in the category of Banach spaces ...
CommunityBot
- 1
6
votes
0
answers
639
views
Hilbert subspaces of indefinite inner product spaces
Let $E$ be a real linear space, endowed with a non-degenerate symmetric
bilinear form $(.,.)$.
Suppose that the [indefinite] inner product space $(E,(.,.))$
satisfies the following [sequential] ...
- 25.8k
12
votes
3
answers
1k
views
What's algebraic approach to QM good for?
The algebraic formulation of quantum mechanics (and related stuff, like quantum thermodynamics & dynamical systems etc.) via C*-algebras provides a viewpoint based mostly on abstract functional ...
- 21.5k
5
votes
2
answers
862
views
Hilbert $C^*$-modules and approximate units
Hi,
Given a $\sigma$-unital $C^*$-algebra $A$ and a full Hilbert $A$-module $E$, is it possible to find an approximate unit $ \{\epsilon_i\}, i\in I$ in $A$ such that each $\epsilon_i$ is of the ...
CommunityBot
- 1
11
votes
3
answers
1k
views
Continuous automorphism groups of normed vector spaces?
Consider the metric space on, say, ℝ2 induced by the various $L^p$ norms, and the group of isometries from that space into itself that preserve the origin. When $p=2$ I get the continuous group ...
- 4,060
8
votes
0
answers
605
views
convergence rate in Wiener's approximation theorem
Wiener has the following fantastic results about approximations using translation families:
Given a function $h: \mathbb{R} \to \mathbb{R}$, the set $\{\sum a_i h(\cdot - x_i): a_i, x_i \in \mathbb{...
- 1,839
10
votes
0
answers
609
views
Asymptotic non-distortion of the separable Hilbert space
By the work of E. Odell and Th. Schlumprecht, we know that the
separable Hilbert space $\ell_2$ is arbitrarily distortable. But
I don't know if an "asymptotic" version of their result is true.
To ...
- 25.8k
4
votes
2
answers
4k
views
Compact Convex sets and Extreme Points
There are examples that show the set of extreme points of a compact convex subset of a locally convex topological vector space need not be closed when the real dimension of the space is at least 3. ...
- 4,060
21
votes
5
answers
4k
views
Isomorphisms of Banach Spaces
Suppose $X$ and $Y$ are Banach spaces whose dual spaces are isometrically isomorphic. It is certainly true that $X$ and $Y$ need not be isometrically isomorphic, but must it be true that there is a ...
- 4,060
20
votes
3
answers
4k
views
What is the origin of the term "spectrum" in mathematics?
The use of the term "spectrum" to denote the prime ideals of a ring originates from the case that the ring is, say, $\mathbb{C}[T]$ where $T$ is a linear operator on a finite-dimensional vector space; ...
- 6,523
2
votes
3
answers
1k
views
Baire category theorem
Let's call the following conditions (1): $X$ is a complete metric space with metric $d$, $X = \cup_{n=1}^\infty A_n$. Let $\bar{A}$ denote the closure of $A$.
Let's call the following statement (2): ...
- 498
8
votes
3
answers
698
views
L_p norm balls for 1<p<2 - is it always similar to an L_q norm ball for some q>2?
The L_1 ball in 2D is shaped like a diamond (L_1 is also known as the Manhattan norm). The L_∞ ball is shaped like a square (L_∞ is also known as the supremum norm). They are similar, i.e. have same ...
- 31.5k
11
votes
2
answers
862
views
Monotone Lipschitz embedding ?
In 1974, Aharoni proved that every separable metric space (X, d) is Lipschitz isomorphic to a subset of the Banach space c_0.
Thus, for some constant L, there is a map K: X --> c_0 that satisfies the ...
- 61.9k
5
votes
0
answers
417
views
Direct integrals and fields of operators
Suppose we have a measure space $(X,\mu)$ and a measurable field of Hilbert spaces $H_x$ on it. We can form the direct integral ${\cal{H}} = \int H_x \ d \mu$, which is a Hilbert space.
Suppose now ...
- 3,897
2
votes
1
answer
168
views
Local supporting points of Lipschitz functions
Let X be a separable reflexive Banach space and f:X\to\mathbb{R} be a
Lipschitz function. Say that a point x in X is a local supporting point
of f if there exist x^* in X^* and an open neighborhood U ...
- 45k
5
votes
1
answer
514
views
Request for reference: Banach-type spaces as algebraic theories.
Sparked by Yemon Choi's answer to Is the category of Banach spaces with contractions an algebraic theory? I've just spent a merry time reading and doing a bit of reference chasing. Imagine my delight ...
CommunityBot
- 1
7
votes
1
answer
570
views
Categorical duals in Banach spaces
Near the bottom of the nlab page for Banach space I see "To be described: duals (p+q=pq)".
Are $(\mathbb{R}^n)_p$ and $(\mathbb{R}^n)_q$ dual objects in the closed symmetric monoidal category of ...
- 25.8k
26
votes
3
answers
2k
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Universality of zeta- and L-functions
Voronin´s Universality Theorem (for the Riemann zeta-Function) according to Wikipedia: Let $U$ be a compact subset of the "critical half-strip" $\{s\in\mathbb{C}:\frac{1}{2}<Re(s)<1\}$ with ...
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3
answers
2k
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What are some interesting sequences of functions for thinking about types of convergence?
I'm thinking about the basic types of convergence for sequences of functions: convergence in measure, almost uniform convergence, convergence in Lp and point wise almost everywhere convergence. I'm ...
4
votes
3
answers
6k
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Advantages of a back-propagation neural network over other function approximation methods
Hello.
Let's say I have a set of input vectors $I = \{\mathbf{x_1}, \dots, \mathbf{x_k}\} \subset \mathcal{R}^m$ and a set of output vectors $O = \{\mathbf{y_1}, \dots, \mathbf{y_k}\} \subset \...
- 2,133
6
votes
1
answer
726
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The "ultimate" indefinite inner product space
This can be considered as a relative of Splitting a space into positive and negative parts.
Is there a real (non-trivial) vector space $V$, endowed with a nondegenerate symmetric bilinear pairing $\...
CommunityBot
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10
votes
1
answer
635
views
What's the nearest algebraic theory to inner product spaces?
Following the references to the accepted answer to Is the category of Banach spaces with contractions an algebraic theory? one discovers that there is an algebraic theory (infinitary) which is closely ...
CommunityBot
- 1
6
votes
2
answers
1k
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Closed, complemented subspaces of $l^1(X)$ when $X$ is uncountable
... are all isomorphic to $l^1$ on some other index set. At least, that much I "know" from 2nd-hand sources, since the original proof is apparently in a paper of Köthe from the 1930s 1960s (in ...
- 31.5k
4
votes
1
answer
321
views
What functorial topologies are there on the space of linear maps between LCTVS?
Setup: we consider the category of locally convex topological vector spaces with morphisms as continuous linear maps. This time, I'm explicitly allowing the axiom of choice (or at least the Hahn-...
- 47.3k
3
votes
3
answers
2k
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Conditional expectation of convolution product equals..
Let $X, Y$ be two $L^1$ random variables on the probablity space $(\Omega, \mathcal{F}, P)$. Let $\mathcal{G} \subset \mathcal{F}$ be a sub-$\sigma$-algebra.
Consider the conditional expectation ...
5
votes
2
answers
765
views
Can we distinguish the algebraic and continuous duals of a Banach space without choice (or HBT)?
The algebraic dual of a normed vector space is the space of all linear functionals to the ground field (either $\mathbb{R}$ or $\mathbb{C}$ for this question). The continuous dual is the subspace of ...
CommunityBot
- 1
4
votes
5
answers
3k
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Generalize Fourier transform to other basis than trigonometric function
The Fourier transform of periodic function $f$ yields a $l^2$-series of the functions coefficients when represented as countable linear combination of $\sin$ and $\cos$ functions.
In how far can this ...
- 25.6k
9
votes
2
answers
1k
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Explicitly describing extreme points of infinite dimensional convex sets
I am currently trying to apply some results from Choquet theory - i.e., the generalisation of results by Minkowski and Krein-Milman for representing points in a compact, convex set C by probability ...
- 9,366
5
votes
3
answers
230
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Is the Fell-Doran problem trivial in a topological setting?
The Fell-Doran problem is a problem in functional analysis. It goes as follows: Let $A$ be a complex unital algebra, $X$ a locally convex space, and $L(X)$ the algebra of all continuous endomorphisms ...
- 26.8k