Math 55a: Norm basics

Let F be either of the fields R and C, and let V be a vector space over F. A norm on V is a real-valued function ||·|| on V satisfying the following axioms:
  1. Positivity: ||0||=0, and ||v|| is a positive real number for all nonzero vectors v.
  2. Homogeneity: ||cv|| = |c| ||v|| for all scalars c and vectors v.
  3. Subadditivity: ||v+w|| <= ||v|| + ||w|| for all vectors v,w in V.
A normed vector space V is automatically a metric space with the distance function d(v,w):=||v-w||. (This still holds if Homogeneity is replaced by the weaker axiom ||v||=||-v||.)

Two norms, say ||·|| and [[·]], on a vector space are said to be equivalent if there exist positive constants C,C' such that ||v||<=C[[v]] and [[v]]<=C'||v|| for all vectors v. Equivalent norms yield the same notions of open/closed/bounded/compact sets, convergence, continuity and uniform continuity, and completeness.

If V is finite-dimensional, all norms on V are equivalent. In particular, the above notions are canonically defined, independent of choices of basis or norm (since we already know that any finite-dimensional F-vector space already has at least one norm). Proof: First check that equivalence of norms is in fact an equivalence relation. It is then enough to fix one norm ||·||, say the sup norm relative to some choice of basis, and show that any other norm is equivalent to it. Using homogeneity and subadditivity we see that [[v]]<=C'||v|| where C' is the sum of the [[·]]-norms of the unit vectors. In particular, it follows that the function [[·]] is continuous in the ||·||-metric. To get the reverse inequality, first use homogeneity to reduce it to the case of ||v||=1. The set of such vectors v is closed and bounded, hence compact by Heine-Borel. The function 1/[[·]] on this set is continuous, because it is the multiplicative inverse of a continuous function to the positive reals. Hence it is bounded. We may use an upper bound for C.

An infinite-dimensional vector space may have inequivalent norms. For example, you can easily check that the sup and sum norms on Foo are not equivalent, and readily construct many more pairwise inequivalent norms on this space.