Definition of inner product space

An inner-product space is a vector space equipped with a specified inner product. Examples include $\mathbb C^n$ with $x^*y$, matrices with $\langle A,B\rangle=\operatorname{tr}(A^*B)$, and continuous functions on $[a,b]$ with $\int_a^b\overline{f(t)}g(t)\,dt$ for $a<b$. Choosing an inner product is additional structure, not determined by vector space axioms alone.Copy