Exercise 10, § 6

a) In a Banach space $F$, let $\mathbf{a}$ and $\mathbf{b}$ be two vectors such that $|\mathbf{a}| = |\mathbf{b}| = 1$. Show that for every number $t$ such that $0 \leq t \leq 1$, and for every $p$ such that $1 \leq p < +\infty$, (1)

(2) (express $\mathbf{a} - t^p\mathbf{b}$ as a linear combination of $\mathbf{a} - t\mathbf{b}$ and $\mathbf{a} - \mathbf{b}$ and note that for $0 \leq p \leq 1$, one has $|\mathbf{a} - p\mathbf{b}| \geq 1 - p$ and $|\mathbf{a} - \mathbf{b}| \leq 2|\mathbf{a} - p\mathbf{b}|$). From this, deduce that if $\mathbf{y}$ and $\mathbf{z}$ are any two vectors in $F$, then (3) (4)

b) Show that the mapping $f \mapsto |f|^{(1/p)-1} \cdot f$ is a uniformly continuous bijective mapping of $\mathscr{L}_F^1$ onto $\mathscr{L}_F^p$ (make use of the inequality (3)).

c) Show that the mapping $f \mapsto |f|^{p-1} \cdot f$ of $\mathscr{L}_F^p$ onto $\mathscr{L}_F^1$ is uniformly continuous on every bounded subset of the space $\mathscr{L}_F^p$ (make use of the inequality (4) and Hölder’s inequality). Deduce from this that the topological spaces $\mathscr{L}_F^1$ and $\mathscr{L}_F^p$ are homeomorphic.