Orthonormal set

Definition

Given an orthogonal set, $S\subset X$, where $X$ is an i.p.s, we say $S$ is orthonormal if:

• $\forall x\in S$ we have $\Vert x\Vert=1$

  (Where $\Vert x\Vert:=\sqrt{\langle x,x\rangle}$)

Recall that to be an orthogonal set we must have:

• $\forall x,y\in S[x\ne y\implies x\perp y]$ where:
  • $x\perp y$ denotes that $x$ and $y$ are perpendicular

Questions

• What about the zero vector, we know that $\forall x\in X[\langle x,0\rangle=0]$

Examples

• Obviously the set $\{(0,0,1),(0,1,0),(1,0,0)\}\subset\mathbb{R}^3$ (in Euclidean $3$-space)
• The set $\{(1,0,0,\cdots),(0,1,0,\cdots),\cdots,(0,0,0,\cdots,0,1,0,\cdots)\}\subset l_2$ (in Space of square-summable sequences)

See also

• Perpendicular