Questions tagged [fa.functional-analysis]

Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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Continuous function sort

If you have a real-valued function f(x), positive, continuous and bounded on some interval, then what kind of transform would convert this to a monotonic function g(x) on that interval analogously to ...
user19172's user avatar
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3 votes
2 answers
743 views

Special values of a doubly periodic meromorphic function

Consider the following function: $G(z) = \prod_{n \in \mathbb{Z}} {1 \over{\tanh^2\left(\pi\left(z-n\right)\right)}}$. By constuction, it has poles at $z=m+in$ with $m,n \in \mathbb{Z}^2$. ...
Aobara's user avatar
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13 votes
2 answers
758 views

Properties of orthogonality-preserving c.p. maps between $C^*$-algebras

Suppose that $A,C$ are $C^*$-algebras and $\phi:A \to C$ is a completely positive, orthogonality-preserving linear map. (Orthogonality preserving means: if $a,b \in A$ satisfy $ab=0$ then $\phi(a)\phi(...
Aaron Tikuisis's user avatar
2 votes
1 answer
352 views

Is it true that $c_0(X)^* = \ell_1(X^*)$ ?

I'm trying to prove this that but I can't . Any help/reference ?
Rafael's user avatar
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1 vote
0 answers
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iterated traces for sobolev functions

It is well known that if $M$ is a smooth $(n-1)$-dimensional surface in $\mathbb R^n$ (e.g. a subspace) then there is a continuous trace operator $W^{s,p}(\mathbb R^n)\to W^{s-1/p,p}(M)$. Now suppose ...
Mircea's user avatar
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5 votes
1 answer
629 views

Are piecewise linear curves dense among Hölder curves?

Consider for some $0 < \alpha \leq 1$ the space functions $x:[0,1] \to \mathbb{R}^n$ such that $x(0) = 0$ and $\sup_{s,t} \frac{\|f(t)-f(s)\|}{|t-s|^{\alpha}}$ is finite. There are at least two ...
Pablo Lessa's user avatar
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1 vote
1 answer
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analytic continuation of a Laplace transform from a countably infinite set of points?

Let $f(\lambda)=\int_0^\infty e^{-\lambda s} F(ds)$, where $F$ is the distribution of a positive random variable. Suppose I know the value of $f(n)$ for $n=0,1,2,\cdots$. Is this enough to uniquely ...
psyduck's user avatar
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7 votes
2 answers
887 views

Exotic spectrum of Laplace operator

Given a closed Riemannian manifold and a generalized Laplace $\Delta$ operator, it is well known that $\Delta$ has discrete spectrum $(\lambda_n)_n$ (arranged in a increasing way, not counting ...
Alex's user avatar
  • 101
2 votes
0 answers
542 views

Young inequality in weighted spaces

Let $U$ be a bounded open set in $\mathbb{R}^2$, $g\in L^1_{\mathrm{loc}}(\mathbb{R}^2)$. Let moreover $w$ be a weight (i.e. a non vanishing locally integrable function) on $U$ and $p\geq2$. Does ...
Samuele's user avatar
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21 votes
2 answers
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Self-dual normed spaces which are not Hilbert spaces

Are there any examples of non-Hilbert normed spaces which are isomorphic (in the norm sense) to their dual spaces? Or, is there any result in Functional Analysis which says that if a space is self-...
Uday's user avatar
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2 votes
2 answers
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A point in the weak closure but not in the weak sequential closure

I'm trying to find a proof of this counterexample by von Neumann: Let $x_{mn}\in \ell^2$ be defined by $$x_{mn}(m)=n \quad,\quad x_{mn}(n)=m \quad,\quad x_{mn}(k)=0 \hbox{ otherwise,} $$ and let $S=\...
Kale's user avatar
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-2 votes
1 answer
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Multiplying two Fourier series gives one Fourier series, but what are the new coefficients? [closed]

If I have $A(x)=B(x) C(x)$ (sine periodic from 0 to 1) rewritten as $\sum_n A_n \sin(n \pi x)=\sum_m B_m \sin(m \pi x)\sum_p C_p \sin(p \pi x)$ is there any easier way to compute $A_n$ from $B_m,...
Lababidi's user avatar
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8 votes
2 answers
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Approximation by polynomials

Let $f:[a,b] \rightarrow \mathbb{R}$ be of class $C^n$. Let $ x_0, ..., x_m$ be different numbers from $[a,b]$. Does for each $\varepsilon >0$ there exist a polynom $P$ such that $P^{(k)}(x_i)=f^{...
arc's user avatar
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4 votes
1 answer
261 views

Exotic uniform algebras

The first non-trivial example of a uniform algebra which comes to mind is the disc algebra $A(\mathbb{D})$. In a similar manner one can define its relatives $P(U)$ and $R(U)$, where $U$ is any region ...
Alex Ortega's user avatar
5 votes
3 answers
788 views

are the smooth vectors of a Frechet space dense?

Given an action $\alpha$ of $V$ a Lie group on $B$ a Fréchet space with seminorms $ \{ \| \cdot \|_j \} $, let $B^\infty$ be the space of smooth vectors. Is this dense in $B$? Can I guarantee it is ...
Yul Otani's user avatar
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7 votes
2 answers
461 views

Extension of weakly compact operators from $\ell_1$ into $c_0$

Is every weakly compact operator from $\ell_1$ into $c_0$ extendible to any larger space? Equivalently, is every weakly compact operator from $\ell_1$ into $c_0$ extendible to $\ell_\infty$?
Joaquin M. Gutierrez's user avatar
7 votes
0 answers
263 views

Problem with Shelah and Stern's paper on the Hanf number of the theory of Banach spaces

I have been trying to understand "The Hanf number of the first order theory of Banach spaces" by Shelah and Stern (Trans. AMS 244 (1978) 147-241). They construct a normed space $M$ from a Hilbert ...
Rob Arthan's user avatar
14 votes
3 answers
1k views

Extreme points of unit ball in tensor product of spaces

Let $B_1, B_2$ be unit balls in finite-dimensional normed spaces $X_1, X_2$ respectively. Let $e(B_1), e(B_2)$ be corresponding extreme points sets. Consider the unit ball $B$ in tensor product $...
Yauhen Radyna's user avatar
18 votes
7 answers
7k views

Grothendieck on Topological Vector Spaces

In a short biography article on Alexander Grothendieck, it is mentioned that after Grothendieck submitted his first thesis on Topological Vector Spaces (TVS), apparently, he told Bernard Malgrange ...
7 votes
3 answers
4k views

Is a semicontinuous real function Borel measurable?

Let $f(x,u): [0,1]^2 \mapsto \mathbb{R}$ be a continuous function. [Q] Is $g(x) = \inf_{u\in [0,1]} f(x,u)$ always Borel measurable? If not, can one find a counter-example? Note that, for any $c$, ...
kenneth's user avatar
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5 votes
1 answer
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Inner product of linear bounded operators between Hilbert spaces

Let $X$ and $Y$ be Hilbert spaces, and let $L(X,Y)$ be the set of bounded linear operators between Hilbert spaces. Can we equip $L(X,Y)$ with a natural inner product? I think it should look like $\...
shuhalo's user avatar
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0 votes
1 answer
216 views

Sort-of extension of Young inequality to arbitrary measures

Hello folks, Suppose we're working in ${\mathbb T}^q$, the $q$-times Cartesian product of $[-\pi,\pi]$ with the endpoints identified. So, $ L^p := L^p({\mathbb T}^q) $. The Young inequality may be ...
Seaking's user avatar
9 votes
2 answers
1k views

polynomials with minimal $L_\infty$ norm on multiple disjoint intervals

It is well-known that Chebyshev polynomials are the polynomials of minimal $L_\infty$ norm on [-1,1] with leading coefficient 1. But what if you want the minimal $L_\infty$ polynomial on two disjoint ...
Paul's user avatar
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1 vote
1 answer
138 views

Estimating norms of derivatives

Let $(M,g)$ be a closed, compact Riemannian manifold. Let $u$ be a smooth function. Let $H^{-k}(M)$,, $k$ is a positive integer, be the dual Hilbert space of $H^{k}(M)$. Does it follow that $|| |\...
Viktor Bundle's user avatar
2 votes
3 answers
3k views

Extension of pointwise convergence of a sequence of uniformly continuous functions that converges on a dense set

It is known that a sequence of continuous functions on a metric space that converges pointwise on a dense subset need not converge pointwise on the full space. But what about if one assumes uniform ...
Joakim Arnlind's user avatar
1 vote
0 answers
177 views

Inequalities between self-adjoint operators

Let $T_s$ ($s\ge0$) be a smooth family of non-negative self-adjoint operators in a separable Hilbert space $H$. Suppose that, for some $C'>C>0$, we have $T_0+Cs^2\le T_s\le T_0+C's^2$ for all $s$...
Jesús Álvarez's user avatar
10 votes
0 answers
497 views

Lacunary hyperbolic groups and weak amenability

In the paper called Lacunary Hyperbolic group, Y. Ol'shanskii, D. Osin and M. Sapir define and characterize the lacunary hyperbolic groups, which contains the hyperbolic groups but also Tarski's ...
Denis Poulin's user avatar
7 votes
1 answer
1k views

weak*-closed subspaces

Recall that a closed subspace $Y$ of a Banach space $X$ is weakly complemented if the set $$Y^{\bot}:= \{ f\in X^*| f(y) = 0 \forall y\in Y\}$$ is a complemented subspace of $ X^*$. For example, $c_0$ ...
Denis Poulin's user avatar
1 vote
2 answers
289 views

The operator preseving two disjoint dense operator ranges invariant

Let $\mathcal{H}$ be a separable infinite dimensional Hilbert space. Suppose that $\mathcal{H}_0\subset\mathcal{H}$ which is a dense proper subspace is the range of some bounded linear operator $T$. ...
Wenming Wu's user avatar
8 votes
2 answers
1k views

When is the norm of all positive operators on an ordered Banach space determined by their values on the positive cone?

I'm trying to investigate the interplay between the norm and cone of positive elements in ordered Banach spaces. In particular, I would like a nice characterization of when the norm of a positive ...
Miek Messerschmidt's user avatar
2 votes
1 answer
2k views

Invariant functionals on C(R) and amenable groups

Since there seems to be no progress in this interesting question, I took the liberty to reformulate it in a way, that is easier to understand. Moreover, my answer shows that the question is related to ...
Mariarty's user avatar
  • 385
23 votes
5 answers
4k views

Understanding/Mastering Analysis in Topology, necessary?

I have spoken to one professor so far about this, which of course was helpful, and so I am looking for additional opinions: To work with topological tools that were built via analysis, should I be a "...
Chris Gerig's user avatar
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6 votes
2 answers
718 views

Transpose of unbounded operators between Banach spaces.

Let $X$ and $Y$ be Banach spaces, and let $L : X \rightarrow Y$ be a unbounded operator with dense domain $\operatorname{dom}(L)$. We can then talk about the transposed operator $L' : \operatorname{...
shuhalo's user avatar
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7 votes
1 answer
1k views

laplace equation on manifolds with boundary

in aubin's book on page 104 theorem 4.7 there is the theorem: Let $(M,g)$ be a compact $C^{\infty}$ Riemannian manifold. There exists a weak solution $\varphi \in H_{1}$ of $\Delta \varphi = f $ if ...
william's user avatar
  • 213
11 votes
4 answers
1k views

Orthogonality in non-inner product spaces

I have come across a notion of orthogonality of two vectors in a normed space not necessarily inner product space. Two vectors $x$ and $y$ in a normed space are said to be orthogonal (represented $x\...
Uday's user avatar
  • 2,209
11 votes
1 answer
923 views

Applications of the "almost commuting" theorem of H. Lin

H. Lin proved that "almost commuting" hermitian matrices are "nearly commuting." To be more precise, Lin showed that given $\epsilon > 0$ there exists a $\delta > 0$ such that if $A, B \in M_N$ ...
Mustafa Said's user avatar
  • 3,679
3 votes
2 answers
986 views

Do these kernel functions satisfy the semigroup property?

Define the kernel functions for $a\ge 1$, $$ G_a(t,x) := \frac{C_a t}{t^{1+1/a}+|x|^{1+a}}, \qquad \forall t>0,\: x\in \mathbb{R}\;, $$ where the constant $C_a$ is some normalization constant ...
Anand's user avatar
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1 vote
0 answers
132 views

Inequality involving BV norm and a regularizing kernel

In the same article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/# (related to this ...
Beni Bogosel's user avatar
  • 2,102
19 votes
3 answers
1k views

Is there "Schur-Weyl duality" for infinite dimensional unitary group?

To what extent does the relation between the diagonal representation of $SU(n)$ in $(\mathbb{C}^n)^{\otimes k}$ and representations of the symmetric group $S_k$ remain valid when instead of the group $...
Michał Oszmaniec's user avatar
14 votes
2 answers
6k views

Are weak and strong convergence of sequences not equivalent?

For some infinite-dimensional Banach spaces $E$, it is easy to find sequences $\langle x_i:i\in\mathbb N_0\rangle$ which converge to zero weakly but not in the norm topology, i.e. we have $\lim_{i\to\...
TaQ's user avatar
  • 3,348
1 vote
1 answer
163 views

Maximum number of orthonormal vectors contained in an open cone

Let $H$ be a separable Hilbert space, $\Pi:H\to L$ the orthogonal projection to a linear subspace of finite dimension $p$, and $U$ the open cone of vectors $u\in H$ such that $\langle u,\Pi u\rangle&...
Jesús Álvarez's user avatar
1 vote
1 answer
278 views

A nice overview (and maybe derivation) of the Poincaré transformations of the Vector Spherical Harmonics

With $Y_{lm}(\vartheta,\varphi)$ being the Spherical Harmonics and $z_l^{(j)}(r)$ being the Spherical Bessel functions ($j=1$), Neumann functions ($j=2$) or Hankel functions ($j=3,4$) defining $$\psi_{...
Tobias Kienzler's user avatar
3 votes
1 answer
331 views

Stronger bound for a modified Lyapunov Equation

In regard to the stability analysis and control properties of the linear system $\dot{x}=Ax$. Consider the solution $P$ of the continuous Lyapunov equation $AP+PA^T+Q=0$, where $A,Q,P \in {\mathbb{R}...
user21598's user avatar
2 votes
0 answers
173 views

A limit involving a regularizing kernel

I am studying the following article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/# ...
Beni Bogosel's user avatar
  • 2,102
3 votes
1 answer
625 views

Converse of the taylor's theorem in Banach Spaces

I would like to known if the following converse of the taylor's theorem is true: Let $E$, $F$ Banach spaces, and $f:E\rightarrow F$ continuous. Suppose there are $k$ continuous functions $T_i: E \...
Ferraiol's user avatar
  • 121
6 votes
0 answers
690 views

What is the structure of a space of $\sigma$-algebras?

Let $X$ be a compact metric space, and consider the Banach space $\Omega = C(X,\mathbb R)$ of continuous, real-valued functions on $X$, equipped with the supremum norm. Let $\delta_x \in \Omega^*$ be ...
Tom LaGatta's user avatar
  • 8,372
5 votes
1 answer
4k views

Contraction mapping with no fixed point

I am interested in constructing the following "counter-example" to the Banach's fixed point theorem. Let $K=$ {$ g\in L_1: \|g\|=1, g(\cdot)\ge0 $}. Clearly, $K$ is not a compact and $K$ is not ...
Oleg's user avatar
  • 911
8 votes
1 answer
475 views

Continuous selections from sums of compact sets

This question is somehow related to the last open problem from Grothendieck's thesis about completeness of locally convex inductive limit. However, a particular case of the problem boils down to a ...
Jochen Wengenroth's user avatar
2 votes
2 answers
773 views

Principle of Local Reflexivity

I'm having a hard time trying to understand a proof of the Principle of Local Reflexivity. I'm following the proofs from 1) Topics in Banach Space Theory (by Fernando Albiac, Nigel J. Kalton) 2) ...
Rafael's user avatar
  • 151
2 votes
0 answers
146 views

Subspace where an operator is positive

Given a self-adjoint operator $\hat{T}$ on a Hilbert space $\mathcal{H}$, and assuming it has a basis of eigenvectors $\{\phi_n\}$ such that $\hat{T}\phi_n=\lambda_n\phi_n$, one can consider the ...
Emilio Pisanty's user avatar

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