# Tagged Questions

27 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 ...
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 ...
24 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 $||f||_{\infty} \leq 1$.
39 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: ...
405 views

### Banach space valued integration (Riemann type)

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: ...
197 views

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 $f ... 1answer 51 views ### Convergence in$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 ... 2answers 118 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 ... 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$x\in X+Y$... 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$0$... 1answer 122 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],$1 ...
Note ($\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 $X\in A$, ...