Inner product

An inner product assigns a scalar $⟨u,v⟩$ to two vectors and supplies their lengths and orthogonality. Using the convention linear in the first argument, it satisfiesThe resulting norm is $\Vert v\Vert =\sqrt{⟨v,v⟩}$. Over the real numbers, conjugation has no effect and the form is symmetric and bilinear. Over the complex numbers, it is sesquilinear; the convention linear in the second argument is also common, so formulas for coefficients must match the stated convention.

The Euclidean example is $⟨u,v⟩={\sum }_i{u}_i\overline{v_i}$. For real $u=(2,1)$ and $v=(1,3)$ this gives $⟨u,v⟩=5$, while $\Vert u{\Vert }^2=5$ and $\Vert v{\Vert }^2=10$. If $r=(-1,2)$, then $⟨r,u⟩=0$: perpendicularity becomes an algebraic equation. The Cauchy-Schwarz inequality gives $|⟨u,v⟩|\le \Vert u\Vert \Vert v\Vert$, and in a real inner product space it makes the angle formula $\cos \theta =⟨u,v⟩/(\Vert u\Vert \Vert v\Vert )$ meaningful for nonzero vectors.

Different inner products express different measurements. The weighted example $⟨u,v⟩=4u_1{v}_1+u_2{v}_2$ on ${\mathbb{R}}^2$ gives $\Vert (x,y){\Vert }^2=4x^2+y^2$, so its unit-length vectors lie on an ellipse. For functions, an integral such as $⟨f,g⟩=\int f\overline{g}$ supplies the geometry of the L2 space. This lets orthogonal projection and least-squares solutions apply to curves as well as coordinate lists.