Holomorphic functional calculus in non-Archimedean geometry

The purpose of this thesis is to provide a complete proof of the holomorphic functional calculus theorem for unital commutative Banach algebras over a non-Archimedean field $K$, as introduced by V. G. Berkovich in the book "Spectral Theory and Analytic Geometry over Non-Archimedean Fields". Omitting some further properties that are proven, it says that there is a way to extend any bounded homomorphism $\phi\colon A\to D$ from a $K$-affinoid algebra $A$ to a unital commutative Banach $K$-algebra $D$ to a homomorphism $\theta_\phi\colon \Gamma(\Sigma_\phi,O_{M(A)})\to \D$, where $\Gamma(\Sigma_\phi,O_{M(A)})$ is the $K$-algebra of $K$-affinoid functions on the spectrum of $\phi$."