Given a real vector space V, an inner product is a bilinear form V*V → R, commonly denoted as (x,y), which satisfies:

(x,y+z) = (x,y)+(x,z); (x,by) = b(x,y); (x+y,z) = (x,z)+(y,z); (ax,y) = a(x,y).
(x,x) ≥ 0, and (x,x)=0 only for x=0.


These properties are enough to prove the Cauchy-Schwarz inequality, from which it follows that ||x||2 = (x,x) is a norm (Cauchy-Schwarz is required for the triangle inequality). Note, however, that not every norm is derived in this way!

For complex vector spaces, the analogue of an inner product is a hermitian product. Confusingly, physicists will use the bra-ket notation (invented for the Dirac formalism     of quantum mechanics), denoting it as <a|b>. The linear functionals y → <a|y> and x → <x|b> (for fixed a and b) are accordingly denoted <a| and |b>.