Bochner spaces and bounded inclusion of value spaces

If $s_n\to u$ almost everywhere are the simple approximants in [F1], the operator-norm bound in [F4] gives $J{s}_n\to Ju$ almost everywhere. The composed approximants are measurable simple functions, and $\mathrm{\parallel }Ju(t){\mathrm{\parallel }}_F\le \mathrm{\parallel }J\mathrm{\parallel }\mathrm{\parallel }u(t){\mathrm{\parallel }}_E$. Integrating the $p$th powers proves the asserted norm bound and independence of the representative. If $Ju=0$ as a class, pointwise injectivity outside its null set gives $u=0$ as a class.

The displayed norm and quotient form a complete real normed space by [F2]. For complex $E$, the same simple functions, measurable classes and norm define the Bochner space over the realification. Completeness of its unchanged metric proves completeness over the complex field as well. Scalar multiplication and addition act on representatives and preserve the equivalence relation and strong measurability; the scalar triangle inequality for the norm is part of the real completeness interface.

By [F5], ${\int }_0^{T}1dt=T$. For $p>1$, Holder with conjugate exponent $p\mathrm{/}(p-1)$ gives ${\int }_0^T\mathrm{\parallel }u(t){\mathrm{\parallel }}_{E}dt\le ({\int }_0^T\mathrm{\parallel }u(t){\mathrm{\parallel }}_E^{p}dt{)}^{1\mathrm{/}p}({\int }_0^{T}1dt{)}^{1-1\mathrm{/}p}=T^{1-1\mathrm{/}p}\mathrm{\parallel }u{\mathrm{\parallel }}_{L^p(E)}$. For $p=1$ the assertion is equality. This proves the final estimate and completes the claims.