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

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0answers
40 views

Question about lim and sup of a function [on hold]

Let f(t,d) be a continuous function in t. Prove that: $\lim\limits_{m\to \infty}\sup\limits_{t\in[t_1,t_2]}f(t,d_m)=\sup\limits_{t\in[t_1,t_2]}\lim\limits_{m\to \infty}f(t,d_m)$ Does anyone have any ...
1
vote
1answer
91 views

Topology on the dual of a Frechet space

If $F$ is a Frechet space, is there any locally convex space topology on the dual $F'$, such that for each local diffeomorphism $f$ from an open subset $U$ of $F$ to $F$, the map $U \times F' ...
0
votes
0answers
32 views

On existence of right units with control of their norms [on hold]

Does, there exist a Banach algebra with a family of right units with norms converging to 1, but without right unit of norm 1?
-2
votes
0answers
50 views

Property of $\limsup$ in $\ell_1$ [on hold]

Let $w$ be any element in $\ell_1$, and $(w_n)$ be a bounded sequence in $\ell_1$ that converge to 0 pointwise. I want to prove ...
-1
votes
0answers
50 views

Basis for the rational functions [on hold]

The rational functions $f(x)$ are given by $f(x) = \frac{P(x)}{Q(x)}$ where $P(x)$ and $Q(x)$ are polynomials. What bases for the rational functions are generally used in numerical analysis?
2
votes
3answers
215 views

Cardinality of $C^*([0,1])$ [on hold]

What is the cardinality of the continuous dual of $C([0,1])$ (the set of continuous functions from $[0,1]\to \mathbb{R}$)?
0
votes
0answers
38 views

Calculating the the ratio of two Dirac delta functions as the limit of the ratio of nasent delta functions? [on hold]

I am in a situation where I find myself with the ratio of Dirac delta functions. Specifically, I find myself with the ratio of the nascent deltas: $\frac{\lim_{\varepsilon \rightarrow ...
-1
votes
0answers
33 views

Closed sets on the product space and operators [on hold]

$H$ is an hilbert space and $C$ is a closed subset of $H\times H$ with the product topology. If $P$ is the projection $P: (x,y) \in F\times F \to y \in F$ do we have that the set $$P(C)= \{ P((x,y)), ...
-1
votes
0answers
45 views

Approximation of bounded continuous functions by Lispschitz bounded functions

Let $H$ be an Hilbert space and $f : H \rightarrow \mathbb{R}$ a continuous and bounded by $M>0$ function. Is it possible to construct a sequence of functions $f_n$ Lipschitz uniformly bounded by ...
-1
votes
1answer
176 views

How to prove this equality in $\ell_1$ [on hold]

Let $w$ be any element in $\ell_1$, and $(w_n)$, $(z_n)$ two bounded sequence in $\ell_1$ that converge to 0 pointwise. Then we have ...
-2
votes
0answers
65 views

Any other operators that may convert agebraic function into transcendental ones [on hold]

As we know,integral may convert or map a rational function or algebraic function into transcendental one,are there any other operators that may convert a rational function or algebraic function into ...
2
votes
1answer
100 views

BMO spaces on the torus

I was reading BMO spaces (John-Nirenberg) on wikipidia http://en.wikipedia.org/wiki/Bounded_mean_oscillation. There they define BMO norm as $$sup_{Q}\frac{1}{Q}\int_Q |u(y) - u_Q|dy$$ where $u_Q$ is ...
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votes
0answers
28 views

Convergence of independent RVs to a constant [on hold]

Let $(X_n)$ be a sequence of independent RVs that converges in probability to some RV $X$. Show that $X$ is constant a.e. My attempt I think I should use Kolmogorov 0-1 theorem but I don't know how. ...
6
votes
2answers
154 views

elementwise functions of positive definite matrix

The fact that the Schur (that is, element wise) product of two positive definite (symmetric) matrices is positive definite immediately implies (using the convexity of the positive semi definite cone) ...
1
vote
0answers
37 views

On $C_0(\Omega)$-module maps from $L_\infty(\Omega,\mu)$ to $L_q(\Omega,\nu)$

Let $\Omega$ be a locally compact space, and $\mu,\nu\in C_0(\Omega)^*$. By $H_{p,q}^{B}$ (resp. $H_{p,q}^{C}$) we denote the Banach space of continuous $B(\Omega)$-module (resp. $C_0(\Omega)$-module) ...
0
votes
0answers
56 views

How $f$ is approximated, in the $L^{p}-$ norm, by a function $f+h$ whose Fourier transform is constant in some nbhd of the point?

Fact.Suppose $f\in L^{1}(\mathbb R), x_{0}\in \mathbb R,$ and $\epsilon >0.$ Then there exists $h\in L^{1}(\mathbb R)$ with $\|h\|_{L^{1}}< \epsilon,$ such that $$\hat{h}(x)= ...
0
votes
1answer
74 views

Densely-defined unbounded operators with large support

Most densely-defined unbounded linear operators on Hilbert spaces have a very large domain. In fact, for a lot of natural operators the intersection of their domains are still dense. Let us consider ...
0
votes
3answers
169 views

Specific Reference? Noncommutative topology and C^* algebras [closed]

I stumbled across this "dictionary for noncommutative topology" http://planetmath.org/noncommutativetopology and I would be very interested in learning more on the subject, particularly I'd like to ...
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votes
0answers
153 views

Translation is continuous in measure space [closed]

Let $\mu\in L^1[T]$ be a positive function and $f\in L_\mu^1[T]\cap L^\infty(T)$, where $T$ is the unit circle and define $g(t)=f(e^{it})$. Is the following function continuous $$R\ni s \to g_s \in ...
7
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0answers
97 views

Without AC, which implications between the different definitions of amenability still hold?

More precisely, I would like to know which implications between the following definitions of amenability of a discrete countable (or even finitely generated) group can be proved to hold with only ZF ...
0
votes
1answer
65 views

Pointwise (in time) convergence in $H^{-1}$ implies pointwise weak convergence in $L^q$, why?

Let $u_n \to u$ in $C^0([0,T];H^{-1}(\Omega))$ and suppose $\lVert u_n \rVert_{L^\infty(0,T;L^\infty(\Omega))} \leq C$ for all $n$. It follows that for almost all $t$, $u_n(t)$ is bounded in ...
4
votes
1answer
137 views

Is it compatible with ZF to assume that every amenable discrete group is finite?

The question is in the title, amenability being understood as the existence of a left-invariant finitely additive probability measure on the group of interest. The case of countable groups is treated ...
0
votes
0answers
48 views

approximation of $L^{1}-$ function in the $L^{1}-$norm in some particular way [closed]

Fact: Suppose $f\in L^{1}(\mathbb R), \gamma_{0}\in \mathbb R, \hat{f}(\gamma_{0})=0, W$ is a some neighbourhood of $\gamma_{0},$ and $\epsilon >0.$ Then there exists $k\in L^{1}(\mathbb R)$ such ...
1
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0answers
119 views

On sequence of functions $(h_n)$ satisfying $\Vert\sum_{n=1}^\infty f * h_n\Vert=\sum_{n=1}^\infty\Vert f*h_n\Vert$ for all $f\in L_1(G)$

Let $(h_n)$ be a sequence of non-zero functions in $L_1(G)$ (where $G$ is a locally compact group) with the property $$ \left\Vert\sum_{n=1}^\infty f * h_n\right\Vert=\sum_{n=1}^\infty\Vert f*h_n\Vert ...
4
votes
0answers
194 views

Why does it seem that $rca=rba$? [closed]

The following paradox has got me stumped. I'm hoping someone can point out the error. Take a locally compact metric space $X$ and define the $C_b(X)$ and $C_0(X)$ as the spaces of continuous ...
2
votes
1answer
73 views

Commutator with Dirichlet Laplacian

Consider the Dirichlet Laplacian $\Delta$ on a compact Riemannian manifold (with boundary). Consider the operator $T = \sqrt{-\Delta}$. My question is: is there any Leibniz/product rule? Can we say, ...
0
votes
0answers
65 views

Equivalent Gaussian measures

Let $\mu$ be a gaussian measure with eigenpair $\{e_k,2^{-k}\}$ and $\nu$ with eigenpair $\{ Te_k,2^{-k}\}$. Here, T is the unitary operator given by $Tx = x - 2\left\langle x,v \right\rangle v$. Let ...
5
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0answers
66 views

Heuristic interpretation of the 'third index' for Besov and Triebel-Lizorkin spaces

For $p,q \in (0,\infty)$ and $s \in \mathbb{R}$, one can define certain function spaces, $B_s^{p,q}(\mathbb{R}^n)$ and $F_s^{p,q}(\mathbb{R}^n)$, the Besov and Triebel-Lizorkin spaces respectively. ...
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0answers
10 views

Resources for learning integral calculations [migrated]

I am willing to learn about integrals . So i wonder is there any systematic book about the topic that goes progressively in difficulty and complexity . My current level is about knowing the basic ...
0
votes
0answers
119 views

Want to show rigorously $\frac{d}{dt}\int_{\Omega}|u(t)|^r = r\langle u_t(t), |u(t)|^{r-2}u(t)\rangle_{H^{-1}(\Omega), H^1(\Omega)}$

We have a bounded domain $\Omega$ of $\mathbb{R}^n$. Let $$u \in L^2((0,T);H^1(\Omega)) \cap H^1((0,T);H^{-1}(\Omega))\cap L^\infty((0,T);L^\infty(\Omega)).$$ I want to show for $r \geq 2$ that ...
3
votes
0answers
66 views

Variational Principle for a System of Differential Equations

I am studying a differential operator of the form $$ L\left(\begin{array}{c} u \\ v \end{array}\right) = -\Delta \left(\begin{array}{c} u \\ v \end{array}\right) + V(x)\left(\begin{array}{c} u \\ v ...
0
votes
0answers
112 views

$L^{p}(\mathbb R)\subset L^{1}(\mathbb R) \ast L^{p}(\mathbb R), (1< p< \infty)$?

Let $\mathbb T$ be a circle group. In 1939, Salem, has shown that, every member of $L^{1}(\mathbb T)$ can written as a product(convolution) some other two members of $L^{1}(\mathbb T),$ that is, ...
1
vote
2answers
137 views

Measures, orthogonal to holomorphic functions

Let $G$ be a domain in $\mathbb{C}^{d}$ and let $H\left(G\right)$ be the space of all holomorphic functions on $G$. My question is how to characterize all such (Radon) measures $\mu$ on $G$, that ...
2
votes
0answers
71 views

Different definitions of Morrey and Campanato Spaces

The book by Giaquinta defines Campanato spaces using the seminorm: $$[u]_{p,\lambda} = \left(\sup_{\substack{{x_0\in\Omega \\ ...
7
votes
1answer
231 views

How much does the absolute value of an operator behave like an absolute value?

Recall that the absolute value of a bounded operator $T$ on a Hilbert space $H$ is the unique positive operator $|T|$ such that $$\||T|x\|=\|Tx\|$$ for all $x\in H$. It can be defined using the ...
3
votes
1answer
145 views

A functional equality

I don't know if this is known, but I was fiddling around with this equality : $$f:(-1,1)\to (-1,1)\quad \text{satisfies}\quad(f(z)+1)^s=\sum_{j=0}^{\infty}\dbinom{s}{j}f(z^k) \quad \forall z\in ...
12
votes
1answer
261 views

Bases for spaces of smooth functions

Let $S$ denote the space of rapidly decreasing sequences, which means sequences $a=(a_k)_{k=1}^\infty$ such that the numbers $p_d(a)=\sup\{k^d|a_k| : 1\leq k<\infty\}$ are finite for all ...
0
votes
1answer
144 views

A question about PDE argument involving monotone convergence theorem and Sobolev space

I'm reading this paper. In it there is the following argument (see page 240). Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. ...
3
votes
0answers
70 views

Is there any weighted sobolev embedding with non-decaying weight

Is there any weighted sobolev embedding like $$(\int_{R^d}(1+|x|)^s |u|^pdx)^{1/p}\leq C||\nabla^a u||_{L^q}$$? Here $s>0$, and for some appropriate $p, q$.
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0answers
52 views

How to Prove the Semi-parametric Representer Theorem

I have also posted this question on MathSE, so if you think it mustn't be here, please let me know, or just delete it. I was thinking that some people here are more appropriate to answer this ...
2
votes
1answer
127 views

is every point of a Berkovich space a Shilov point?

Let $k$ be an algebraically closed non-Archimedean valued field with the value group $\mathbb R$, and let $X$ be a variety over $k$. Is it true that for any point $x \in X^{an}$ of the Berkovich ...
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0answers
61 views

$L^p$ norm of solution to porous medium equation decreases in time: how to make formal calculation rigorous?

Let $u \in C^0([0,\infty);L^1(M)) \cap W^{1,1}_{\text{loc}}((0,\infty);L^1(M))$ with $u(t) \in H^1(M)$ for a.e. $t$ be the solution of the porous medium equation $\dot u = \Delta (u^m)$ on a compact ...
0
votes
0answers
74 views

elliptic regularity when right hand side in weak $L^p$

I am interested in the following question (whose answer i assume is well known) but just not by me. Suppose $u,f$ are smooth functions defined on $B_1$ and $ \Delta u = f$ in $B_1$ with $u=0$ on $ ...
3
votes
1answer
66 views

Regular commutative Banach algebras which are not closed under complex conjugate

Let $A$ be a semisimple commutative Banach algebra with the maximal ideal space $X$. Further, assume that $A$ is regular i.e. for every closed set $E\subseteq X$ and $x\in X\setminus E$, there is some ...
2
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0answers
108 views

Clarkson's inequalities for Banach space valued functions

In standard analysis, Clarkson's inequalities expresses the norms of the sum and difference of two functions in $L^p$ in terms of the norms of the individual functions. In particular, one may use the ...
0
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0answers
42 views

How to find the uniform convexity of a Banach space?

Consider the Hilbert space $\textit{l}_2$. Define $\|x\|_{\lambda} = \lambda\|x\|_{\infty} + \|x\|_2$, where $\lambda > 0$. Is $\textit{l}_2$ a uniformly convex Banach space with respect to ...
2
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1answer
62 views

Existence of normal structure in strictly convex Banach spaces

Does there exists a strictly convex Banach space which is not uniformly convex and has normal structure ?
0
votes
1answer
85 views

Decay of weak solutions to degenerate parabolic PDEs on manifolds without boundary

I'm interested in degenerate parabolic equations posed on compact manifolds without boundaries and in particular decay estimates of the weak solution of such equations of the form $$|u(t)|_{L^p} \leq ...
2
votes
0answers
165 views

Measurability of a map that takes a functional to its composition with a linear operator

Let $(X,\Sigma_X)$ be a measurable space such that $\Sigma_X$ is countably generated. Let $B_b(X)$ be the Banach space of all bounded $\Sigma_X$-measurable functions $X\to\mathbb{R}$ equipped with ...
3
votes
1answer
140 views

Extension of a bilinear functional

Does any one know an example of a bilinear functional $B:C(X)\times C(Y)\to {\bf R}$ ($X$ and $Y$ are open subsets of Euclid spaces) which cannot be extended continuously to a measure $\mu:C(X\times ...