Analysis II | Part IB, 2001

Define what is meant by a norm on a real vector space.

(a) Prove that two norms on a vector space (not necessarily finite-dimensional) give rise to equivalent metrics if and only if they are Lipschitz equivalent.

(b) Prove that if the vector space VV has an inner product, then for all x,yVx, y \in V,


in the induced norm.

Hence show that the norm on R2\mathbb{R}^{2} defined by x=max(x1,x2)\|x\|=\max \left(\left|x_{1}\right|,\left|x_{2}\right|\right), where x=(x1,x2)x=\left(x_{1}, x_{2}\right) \in R2\mathbb{R}^{2}, cannot be induced by an inner product.

Typos? Please submit corrections to this page on GitHub.