Originally Posted by

**matt33** Hi,

I can? can't? figure out how to solve this problem, although it does not seem very hard...

Let H denote a Hilbert space, and L(H) the vector space of all bounded linear operators on H. Given T in L(H), define the operator norm :

||T|| = inf {B : ||Tv|| < B.||v||, for all v in H}.

a/ Show that ||T+T'|| ≤ ||T|| + ||T'|| whenever T and T' are in L(H). (Should be a weak inequality.)

b/ Prove that d(T,T') = ||T-T'|| defines a metric on L(H).

c/ Show that L(H) is complete in the metric d.