Tagged Questions

1
vote
1answer
39 views

$L_p$ spaces and convergence

The Riesz-Fischer Theorem implies that Lp-convergence implies pointwise a.e. convergence of a subsequence. There is an example that shows that the converse may not be true... Let E = [0, 1], $1 ...
1
vote
0answers
60 views

(Real Analysis) Integration of two functions

Note ($\Omega,A,\mu)$ is a finite additive space. Let $f\in \bar S(A, \mathbb{R})$ and $y\in Y$ where $Y$ is Banach space. Prove $\int _\Omega yf d\mu = y \int _\Omega f d\mu$. Also, for any $X\in A$, ...
1
vote
1answer
67 views

Tensoring subspaces

Let $X$ be a Banach space, $E\subset X$, be a subspace and let $\hat{\otimes}$ denote the projective tensor product. Denote $L_1 = L_1 [0,1]$. Does $E\hat{\otimes} L_1$ embed into $X \hat{\otimes} ...
12
votes
2answers
520 views

isomorphic embedding of $L^{p}(\Omega)$ into $L^{p}(\Omega \times \Omega)$?

Let $(\Omega,\mu)$ be a finite measure space such that $\mu(\Omega)=1$. Suppose $1\leq p \leq \infty$. Let $\psi \colon L^p(\Omega) \to L^p(\Omega \times \Omega)$ be the map which maps $f$ onto the ...
2
votes
1answer
56 views

$a\mapsto \log\left(\lVert f\lVert_{1/a}\right)$ is a convex map

Proof for $a\in (0,1)$ that $a\to F(a)= \log\left(\lVert f\lVert_{1/a}\right)$ is a convex map, if $f\in L^p(X)$ for all $p\geq1$ in some measure space $(X,\mu)$. I'd like to prove that for all ...
0
votes
0answers
31 views

boundedness of the Hilbert transform on Bochner spaces

Let $X$ be a Banach space. Let $H$ be the Hilbert transform (on $\mathbb{R}$ or $\mathbb{T}$). Suppose $1<p,q<\infty$. It is well-known that if $H \otimes Id_X$ is bounded on the Bochner space ...
3
votes
0answers
93 views

What is the Dunford Integral and why is it useful?

Wikipedia defines the Pettis Integral for Banach space valued functions on a measure space by duality. Apparently there is a Dunford integral which specializes to the Pettis integral. What is its ...
0
votes
1answer
57 views

reference for conditional expectation

Suppose $1\leq p<\infty$. Let $E$ be a Banach space. Consider a filtration $F_n$ on some probability space $\Omega$. Let $X\in L^p(\Omega,E)$ where $L^p(\Omega,E)$ denote the Bochner space. In ...
10
votes
1answer
317 views

Proof of Hölder inequality by differentiation

I need a reference where we can read a proof of the inequality $\|f\|_r\leq \|f\|_p^{1-\theta}\|f\|_q^\theta$ where $\frac{1}{r}=\frac{1-\theta}{p}+\frac{\theta}{q}$ for $L^p$-spaces of a measure ...