What does $L^p(A,B)$ mean?

Question

The Lebesgue space or $L^p$ space on a measure space $(\Omega,\Sigma,\mu)$ is the set of functions $f$ satisfying $$\|f\|_p = \left(\int_{\Omega} |f|^p \ d\mu\right)^{1/p}\lt\infty,$$ for $1\le p\lt\infty.$ Now, I recently saw in this question the notation $L^p(A,B)$ where $A$ and $B$ are domains. My guess is $$L^p(A,B) = L^p(A)\cup L^p(B).$$ Is this correct? I chose the $L^p$ space as apposed to continuous functions for variety’s sake. I presume that it has the same meaning across different spaces.

Answer 1

The notation $g\in C^0(B,\mathbb R^3)$ means that the map $g:B\to \mathbb{R}^3$ is continuous. Thus, $B$ is the domain of $g$ and $\mathbb{R}^3$, its codomain.

So in your case, if $f\in L^p(\Omega,\mathbb{C})$, then $f:\Omega\to\mathbb{C}$ is a complex-measurable function such that $$\int_{\Omega} |f|^p \ d\mu\lt\infty.$$

A similar notation is $L^p(\Omega;\mathbb{C})$.

In functional analysis, $B(H,H)$ is the space of bounded linear functionals from $H\to H$, where $H$ is a Hilbert space. This can be abbreviated as $B(H)$.