inner product (thing) by ariels

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

bilinearity: (x,y+z) = (x,y)+(x,z); (x,by) = b(x,y); (x+y,z) = (x,z)+(y,z); (ax,y) = a(x,y).
symmetry: (x,y)=(y,x).
nonnegativity: (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>.

dot product	Cauchy-Schwarz inequality	vector space	hermitian
bilinear form	The mathematician's lightbulb quandary	Riesz Representation Theorem	orthonormal
Positive-Definite Non-Degenerate Symmetric Bilinear Forms	⋅	orthogonal projection	orthonormal matrix
linear functional	Gershgorin's theorem	E2 Clique Application Form	The Dirac formalism of quantum mechanics
metric tensor	rank nullity theorem	orthogonal	Hilbert space
norm	Stieltjes integral	Clifford algebra	orthogonal polynomial