Complex integration in a Banach space.

Question

Let $X$ be a Banach space, and $f:\mathbb{C}\to X$ a function from the complex plan $\mathbb{C}$ with values in $X$. I am trying to give meaning to the integral $$\int_C f(\lambda)\,d\lambda.$$ over a contour $C$.

If $X=\mathbb{C}$ we know that $$\int_C f(\lambda)\,d\lambda=\int_\alpha^\beta f(\phi(t))\phi'(t) \,dt.$$ Where $\phi(t)=x(t)+iy(t),\, \alpha\leq t \leq \beta$ is a parametric representation of the curve $C$.

There's a nice description in Rudin's functional analysis on vector valued integration. It's in chapter 3 — Nick Castillo, Jun 08 '22 at 18:41
Still the same thing, $\int_Cf(\lambda),d\lambda:=\int_{\alpha}^{\beta}f(\phi(t))\phi'(t),dt$. The only thing is you now consider the function $F:[\alpha,\beta]\to X$, $F(t)=f(\phi(t))\phi'(t)$, so you need a notion of integration of functions defined on a measure space (here $[\alpha,\beta]$ with usual Lebesgue measure) taking values in a Banach space (Lang, Amann and Escher, Rudin all treat this sort of stuff; Folland gives a guided exercise for the basic definitions, and I'm sure there are several other references). — peek-a-boo, Jun 08 '22 at 18:50
@peek-a-boo Are the results in the classical complex analysis still true for this generalisation, such as Cauchy-Riemann equations imply analytic, and Cauchy integral formula... — Hidda Walid, Jun 08 '22 at 21:00
Yes, lot's of results extend almost verbatim. See the following answers of mine: Generalizing the Analyticity of holomorphic functions, and Reference request : Holomorphic functions with values in Banach spaces, and here's some stuff about the inverse/implicit function theorem. — peek-a-boo, Jun 08 '22 at 21:08
The unique thing that you need to make the integral well-defined is that $f$ is almost-separable valued. See here — , Jun 08 '22 at 21:52

Complex integration in a Banach space.

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