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Mathematics of Quantization and Quantum Fields by Christian Gérard, Jan Derezinski

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1

Vector spaces

In this chapter we fix our terminology and notation, mostly related to (real and complex) linear algebra. We will consider only algebraic properties. Infinite-dimensional vector spaces will not be equipped with any topology.

Let us stress that using precise terminology and notation concerning linear algebra is very useful in describing various aspects of quantization and quantum fields. Even though the material of this chapter is elementary, the terminology and notation introduced in this chapter will play an important role throughout our work. In particular we should draw the reader’s attention to the notion of the complex conjugate space (Subsect. 1.2.3), and of the holomorphic and anti-holomorphic subspaces (Subsect. 1.3.6 ...

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