Orthonormal set

Definition

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

• [ilmath]\forall x\in S[/ilmath] we have [ilmath]\Vert x\Vert=1[/ilmath]

  (Where [ilmath]\Vert x\Vert:=\sqrt{\langle x,x\rangle}[/ilmath])

Recall that to be an orthogonal set we must have:

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

Questions

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

Examples

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

See also

• Perpendicular