In this chapter we discuss the linear algebra of symplectic vector spaces and symplectic vector bundles. To prepare the ground for the discussion of Künneth structures on manifolds in later chapters we introduce linear Künneth structures on vector bundles, and we work out consequences of the existence of Künneth structures in terms of characteristic classes.

The earlier parts of this chapter contain standard material that some readers may be able to skip. There is a substantial overlap, for example, with Chapter 2 of the book of McDuff-Salamon [McS-95]. The later parts contain some important results that are used throughout the book. While not original, these results clarify some of the folklore revolving around symplectic vector bundles and their Lagrangian subbundles. Our reference for the theory of characteristic classes is Milnor-Stasheff [MS-74].