Orthogonality

Orthogonality is the relation in an inner product space $(V,\langle\cdot,\cdot\rangle)$ defined by

Orthogonality provides a geometric notion of “perpendicularity” that is compatible with the norm coming from the inner product . Fundamental inequalities such as the Cauchy–Schwarz inequality control how orthogonality interacts with lengths.

Examples:

• In $\mathbb{R}^2$ with the standard inner product, $(1,1)$ is orthogonal to $(1,-1)$.
• In $\mathbb{R}^n$, distinct standard basis vectors $e_i$ and $e_j$ are orthogonal when $i\ne j$.
• For periodic functions on $[0,2\pi]$ with $\langle f,g\rangle=\int_0^{2\pi} f(t)g(t)\,dt$, the functions $\sin t$ and $\cos t$ are orthogonal.