Linear Isometry Preserves Inner Product

Theorem

Given an inner product space V where the inner product , induces a norm x=x,x which in turn induces a metric d(x,y)=xy, any linear isometry τ:VV under this metric preserves the inner product, that is:

x=τ(x)xVτ(x),τ(y)=x,y.
Proof

Note that the first line in this proof is one of the polarisation identities:

x,y=12(x2+y2xy2)=12(d(x,0)2+d(y,0)2d(x,y)2)=12(d(τ(x),0)2+d(τ(y),0)2d(τ(x),τ(y))2)=12(τ(x)2+τ(y)2τ(x)τ(y)2)=τ(x),τ(y)

noting that τ(0)=0 by linearity.