Complex Inner Product Space

linear-algebra

Definition

Complex Inner Product Space

A complex inner product space is an inner product space whose underlying vector space is defined over the complex numbers $\mathbb{C}$. Equivalently, it is a complex vector space $V$ equipped with an inner product

$$⟨\cdot ,\cdot ⟩:V\times V\to \mathbb{C}$$

satisfying the usual axioms of an inner product space. In particular, the codomain is $\mathbb{C}$, so conjugation appears in the symmetry law.

As for every inner product space, the inner product induces a norm by

$$\Vert v\Vert =\sqrt{⟨v,v⟩}.$$

Example: The space ${\mathbb{C}}^n$ becomes a complex inner product space with the standard inner product

$$⟨x,y⟩=\sum _{i=1}^n{x}_i\overline{y_i}.$$