Cauchy-Lipschitz (or Picard-Lindelöf) theorem for Banach spaces?

Question

Usually we meet Cauchy-Lipschitz (or Picard-Lindelöf) theorem while solving ODEs in $\mathbb R^n$. However, now I want to apply this theorem to solve a special evolutionary PDE. I look it up in wikipedia, where the formulation is given for an open subset of a Banach space. However, when I go into the proof, I find that, just as ordinary proofs in $\mathbb R^n$ cases, they do integrals, which means vector-valued integrals here. I don't know which kind of vector-valued integral is appropriate in this setting (I only skimmed something on vector-valued integral from Rudin's Functional Analysis). I need a clarification for this, and an accessible reference (friendly to layman) for this.

Any help is welcome, thanks!

Answer 1

If you want to avoid the measure theoretic framework, there is a quick and easy way to build a useful integral for Banach-valued functions. It's called the regulated integral / Cauchy integral. A good reference is Dieudonne's Foundations of Modern Analysis, but I think you can also find it in Bourbaki. In any case, it avoids measure theory and gives an integral that's strong enough for proving Cauchy-Lipschitz in Banach spaces.

Answer 2

For $(X,\mathcal{T},\mu)$ a measure space and $E$ a (complex or real) Banach space, you can define the integral of a function $f$ defined from $X$ to $E$.

For this, you can use the space $\mathcal {E}(X, \mathcal{T};E)$ of simple functions defined from $X$ to $E$. A function $f : X \to E$ defined almost everywhere is said to be $\mu$-integrable (or just integrable), if it exists a sequence $(f_n)$ of functions from $\mathcal {E}(X, \mathcal{T};E)$ converging almost everywhere to $f$ such that $$\lim\limits_{n \to \infty} \int \Vert f-f_n \Vert d\mu =0.$$

As you provided a link to the French wikipedia page, a good French reference is Dérivation, Intégration from Claude Wagschal.

Coming back to ODE, take care that some theorems are not valid for infinite dimensional spaces. In particular, Peano existence theorem might not hold. You have a counterexample here.