##
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:
- Positivity: ||0||=0, and ||
*v*|| is a positive real number
for all nonzero vectors *v*.
- Homogeneity: ||
*cv*|| = |c| ||*v*||
for all scalars *c* and vectors *v*.
- 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
*F*^{oo} are not equivalent, and readily construct
many more pairwise inequivalent norms on this space.