2
votes
2answers
42 views

Banach Space: Open Unit Ball Totally Bounded?

Just to be sure: In an infinite dimensional Banach space the open unit ball cannot be totally bounded, right? The context is that I need this in order to find a lack in here...
17
votes
1answer
415 views

Reinventing The Wheel - Part 2: The Lebesgue Integral

Disclaimer After struggling for some time to find an appropriate definition for the notion of integration I came across another attempt for which I would need your help deciding to what extend this ...
1
vote
0answers
23 views

Helffer-Sjöstrand-Formula: Idea behind?

I have to present the Helffer-Sjöstrand-Formula. Now I'm wondering: Why does it include a factor $\chi(y\langle x\rangle^{-1})$ for some bump function $\chi$ and the chinese symbol ...
1
vote
1answer
32 views

Finite Measure Space: Integral Closure = Bochner Integral

I can't sleep for so long time as the integral gives me headaches. I was looking for so many approaches. Now another one. Hope this works... Let $\Omega$ be a finite measure space and $E$ a Banach ...
2
votes
1answer
51 views

Simple Functions: Uniform Convergence

In the proof to proposition 4.2 of 'The Riemann Integral' it is stated that the net of simple functions converges uniformly for continuous functions. This question aims to prove this in a general ...
2
votes
1answer
32 views

Exercise on L^p spaces

Let $f$ be a function of $L^p([0,2]) \>\> \forall p \in [1, \infty )$ and suppose $||f||_p \leq 1$. Show that $f$ belongs to $L^{\infty}([0,2])$ and $||f||_{\infty} \leq 1$.
1
vote
0answers
40 views

Holomorphic Functional Calculus

Framework: Consider a Banach space: $$(E,\|\cdot\|)$$ Given an unbounded operator: $$T:\mathcal{D}(T)\to E\qquad\mathcal{D}(T)\subseteq E$$ together with its resolvent map: ...
8
votes
2answers
458 views

Reinventing The Wheel - Part 1: The Riemann Integral [closed]

Preface The core of any notion of integral is some sort of weighted sum: $$\sum b\mu(A)$$ Depending on wether the domain or range is decomposed these split into Riemann and Lebesgue type ones: ...
7
votes
2answers
202 views

Integration in Banach spaces - interesting, nice and non-trivial examples needed

I am interested in $\textbf{Integration in Banach spaces}$. Here is a little motivation for my question: Let $\left(X,\|\cdot\| \right)$ be a Banach space, $a,b \in \mathbb{R}$ with $a<b$ and $f ...
3
votes
1answer
51 views

Convergence in $L^1$ of a sequence of functions

I have to see if the following sequence of functions is convergent in the space $L^1[(0,\infty)]$ $$f_n(x)= n\frac{\exp\left(-\frac{n}{2x^2}\right)}{x^3}$$ By definition, $f_n(x)$ is convergent in ...
4
votes
2answers
121 views

Banach space integral via defining it in $X^{**}$ and then proving it's in $X$

Vector-valued integration is something I generally try not to think about very much. I have the impression that it can be a sort of "rabbit hole" of a subtlety if one allows it to be. So, I tend to ...
1
vote
1answer
28 views

$K(t,x)=\inf_{x=a+b,\ a\in X,\ b\in Y}\{\|a\|_X+t\|b\|_Y\}$ and $\int_0^\infty \frac{|K(t,x)|^p}{t}dt<\infty$ implies $x=0$?

Let $X,Y$ be two Banach spaces with respective norms $\|\cdot\|_X$ and $\|\cdot\|_Y$. Suppose that $X$ and $Y$ are subsets of a vector space $Z$. Define $K(t,x)$ for $t\in (0,\infty)$ and $x\in X+Y$ ...
2
votes
1answer
78 views

$F_N(t)=\int_{-1/N}^0\overline{f}(t+h)dh$ implies $\|F_N\|_{L^2((0,T); X)}\leq\|f\|_{L^2((0,T); X)}$?

Let $f\in L^2((0,T); X)$ where $X$ is a Banach space and $0<T<\infty$. Define $F_N: \mathbb{R}\to X$ by $$F_N(t)=\int_{-1/N}^0\overline{f}(t+h)dh$$ where $\overline{f}$ is the extension by $0$ ...
1
vote
1answer
126 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
73 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 $X\in A$, ...
2
votes
1answer
78 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
588 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
58 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 ...
3
votes
0answers
136 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
69 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
364 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 ...