Quadratic map over the real numbers

A quadratic map between real vector spaces $V,W$ is a map $Q:V\to W$ of the form $Q(v)=B(v,v)$ , where $B:V\times V\to W$ is symmetric and bilinear. Then

Q(v+h)=Q(v)+2B(v,h)+Q(h),\qquad Q(tv)=t^2Q(v).

Conversely, the associated bilinear map is recovered by polarization:

B(v,w)=\tfrac12[Q(v+w)-Q(v)-Q(w)].

Linearization

For finite-dimensional spaces, or for a bounded bilinear map between normed spaces, the derivative is $DQ(v)[h]=2B(v,h)$ , and the exact remainder is $Q(h)$ . Mere degree-two homogeneity without the bilinear/polarization property is not the definition used here.

References

Sheldon Axler, Linear Algebra Done Right, 4th ed., Chapters 2–3, 6 and 9.