Can anybody please tell me where can I find the proper definitions for the following spaces? I have been looking on the internet and in other books and have not found clear definitions for all these concepts. They are: $AC\left( % \left[ a,b\right] ;X\right) ;$ $W^{1,p}\left( \left[ a,b\right] ;X\right) ;$ $L^{p}\left( a,b;X\right)$, where $X$ is a Banach space.
What I found is a definition for $L^{p}\left( a,b;X\right) $ that goes like this:
Let $1\leq p\leq \infty .$ We will denote by $L^{p}\left( a,b;X\right) $ the space of measurable functions $u:\left[ a,b\right] \rightarrow X,$ that have the property that \begin{equation*} \left\Vert u\right\Vert _{L^{p}\left( a,b;X\right) }=\left( \int_{a}^{b}\left\Vert u\left( t\right) \right\Vert _{X}^{p}\,dt\right) ^{% \frac{1}{p}}<\infty . \end{equation*}
And I also found a definition for $W^{1,p}\left( \left[ a,b\right] ;X\right)$ that goes like this: \begin{equation*} W^{1,p}\left( \left[ a,b\right] ;X\right) =\left\{ u\in AC\left( \left[ a,b% \right] ;X\right) ;\frac{du}{dt}\in L^{p}\left( \left[ a,b\right] ;X\right) \right\} . \end{equation*}
Does anyone have a different definition for $L^{p}\left( a,b;X\right) $ and what about the other spaces?