This is a short introduction to the basics of the theory of normed tensor products. The m-fold tensor product of linear spaces is defined through the universal property. If the involved spaces are normed, then the projective and injective norms on the tensor product are. Basic properties are given: the metric mapping property and their relationship with continuous linear mappings. The symmetric m-fold tensor product and the symmetric projective and injective norms are defined analogously. These are related to the m-homogeneous polynomials.

We give an introduction to vector-valued holomorphic functions in Banach spaces, defined through Fréchet differentiability. Every function defined on a Reinhardt domain of a finite-dimensional Banach space is analytic, i.e. can be represented by a monomial series expansion, where the family of coefficients is given through a Cauchy integral formula. Every separate holomorphic (holomorphic on each variable) function is holomorphic. This is Hartogs’ theorem, which is proved using Leja’s polynomial lemma. For infinite-dimensional spaces, homogeneous polynomials are defined as the diagonal of multilinear mappings. A function is holomorphic if and only if it is Gâteaux holomorphic and continuous, if and only if it has representation as a series of homogeneous polynomials (known as Taylor expansion). A function is weak holomorphic if the composition with every functional is holomorphic. A function is holomorphic if and only if it is weak holomorphic. Analytic functions are holomorphic.