The seminar will consist of three major parts. First of all, we shall look at tensor products of vector spaces. Here our approach will be to view tensor products as functionals on the space of bilinear forms on the Cartesian product of two vector spaces or as being bilinear forms themselves or, for that matter, as linear operators mapping one vector space on another.

After this introductory part, we shall consider tensor products of Banach spaces. To do so we shall need to define a norm on the tensor products. There is a myriad of what is known as reasonable cross norms for tensor products, each of them making the set of tensor products into a rather neat representation of a certain, otherwise somewhat abstract, class of operators transforming one Banach space into another.

In the second part of the seminar, I shall give an overview of the so-called projective cross norm. In doing so I shall show, among other things, that this norm combines well with quotient operators, the fact from which the projective norm derives its name. We shall also see a few, I believe, interesting specific examples of the projective tensor products, namely the tensor products of which one of the factors is the space l₁ of summable sequences or the space L₁(μ) of integrable functions on a finite measure space (Ω, Σ, μ).

In the third and last part of the seminar, I shall discuss the so-called injective cross norm, which is, in many regards, quite opposite to the projective cross norm. One of the things we shall see is that the injective tensor products respect, so to speak, subspaces, rather than quotients. Finally we shall look at several examples of injective tensor products, namely those of which one of the factors is the space c₀ of sequences converging to zero or the space l₁ of summable sequences or the space C(K) of continuous functions on a compact topological space K or the space L₁(μ) of integrable functions on a finite measure space (Ω,Σ, μ).

https://univienna.zoom.us/j/67922750549?pwd=Ulh5L1QxNFhBOC9PUjlVdG9hc0tmUT09