Vector Spaces
Definition
A vector space \(\mathcal{V}\) over a field \(\mathbb{F}\) is a set of elements (vectors) {|$v$⟩} closed under the action of two maps:
- \(+: \mathcal{V}\times\mathcal{V}\to\mathcal{V}\) (Commutative, Associative, Identity \((|0\rangle)\), Inverse \((-|\cdot\rangle)\) )
- \(*: \mathbb{F}\times\mathcal{V}\to\mathcal{V}\) (Compatibility, Distributivity)
\(\mathbb{F}\) can be \(\mathbb{R}, \mathbb{C}\) or others.
Notation
- Mathematics: \(v\in\mathcal{V}\).
- Physics: \(\vec{v}\in\mathcal{V}\), \(||\vec{v}||=v\).
- We will use: \(|v\rangle\in\mathcal{V}\)