## 0.2 Vector spaces

Following Peano, we define a **vector space** as a mathematical structure in which the notion of linear combination “makes sense”.

More formally, a **complex vector space** is a set **vectors** ^{2}

A **subspace** of **ket** vectors, or simply **kets**.
(We will deal with “bras” in a moment).
A **basis** in *exactly* one way) as a linear combination of the basis vectors; **dimension** of

In fact, this is the space we will use most of the time.
Throughout the course we will deal only with vector spaces of *finite* dimensions.
This is sufficient for all our purposes and we will avoid many mathematical subtleties associated with infinite dimensional spaces, for which we would need to tools of **functional analysis**.

As we said, there are certain “nice properties” that these things must satisfy. Addition of vectors must be commutative and associative, with an identity (the zero vector, which will always be written as

\mathbf{0} ) and an inverse for eachv (written as-v ). Multiplication by complex numbers must obey the two distributive laws:(\alpha+\beta)v = \alpha v+\beta v and\alpha (v+w) = \alpha v+\alpha w .↩︎