# Orthogonality

Definition (Orthogonal Vectors)

Two vectors $x$ and $y$ of an inner product space $(X,⟨⋅,⋅⟩)$ are said to be

orthogonalorperpendicularif $⟨x,y⟩=0$, and this fact is indicated by writing $x⊥y$.

A set $A⊂X$ is called orthogonal, if the elements of $S$ are pairwise orthogonal to each other.

Pythagoras’ Theorem

If $x$ and $y$ are orthogonal vectors of an inner product space, then

$∥x+y∥_{2}=∥x∥_{2}+∥y∥_{2}.$More generally,if ${x_{1},…,x_{n}}$ is an orthogonal set in an inner product space, then

$∥x_{1}+⋯+x_{n}∥_{2}=∥x_{1}∥_{2}+⋯+∥x_{n}∥_{2}.$

The converse of Pythagoras’ Theorem is true for real inner product space, but false in the complex case. For example, let $x$ be any unit vector in a complex inner product space. Then $x$ is not orthogonal to $ix$, since $⟨x,ix⟩=i=0$. However, $∥x+ix∥_{2}=∣1+i∣_{2}=2=1+1=∥x∥_{2}+∥ix∥_{2}$.

Definition (Orthonormal Set)

A subset $S$ of an inner product space is called

$⟨x,y⟩={01 x=y,x=y. $orthonormalif we have for all $x,y∈S$

In other words, an orthonormal set is an orthogonal set of unit vectors.

Every orthonormal set is linearly independent.

Proof Suppose that ${x_{1},…,x_{n}}$ is a finite subset of $S$ and that

$α_{1}x_{1}+⋯+α_{n}x_{n}=0$for some scalars $α_{1},…,α_{n}$. Application of $⟨x_{i},⋅⟩$ yields $α_{i}=0$ for all $i∈{1,…,n}$. $□$

Recall that a subset $S$ of a normed space $X$ is called total if its span is dense in $X$.

Definition (Orthonormal Basis)

A total orthonormal set in an inner product space is called

orthonormal basis(orcomplete orthonormal system).

Every Hilbert space has an orthonormal basis.