Complex dot product

This section is about the definition of the dot product over ${\mathbb{C}}^n$, which extends the definition of the dot product over ${\mathbb{R}}^n$.

Some motivation is discussed at: math.stackexchange.com/questions/2459814/what-is-the-dot-product-of-complex-vectors/4300169#4300169

The complex dot product is defined as:

E.g. in ${\mathbb{C}}^1$:

We can see therefore that this is a form, and a positive definite because:

Just like the usual dot product, this will be a positive definite symmetric bilinear form by definition.