Inner Product on a Vector Space

Let V 𝑉 V be a Vector Space over some Field k 𝑘 k that is equipped with an Involution ( − ) ∗ : k → k : superscript → 𝑘 𝑘 (-)^{*}:k\to k . An inner product on V 𝑉 V is a function ⟨ − , − ⟩ : V × V → k : → 𝑉 𝑉 𝑘 \langle-,-\rangle:V\times V\to k that is Hermitian Form.

Examples

The canonical example of an inner product is the Dot Product on the vector space k n superscript 𝑘 𝑛 k^{n} .