suppose that mu(X)<infinity and let p,q belongs to [1,infinity) ,p<q.

  use Holder's Inequality to prove the following:

 1) L q space is subset of L p space.

  2)!!f!!p is less than or equal = mu(X)to the power 1/p - 1/q  !!f!!q
for  all f belongs to L p space.

  3) convergence in L q space implies convergence in L p space.

  and also show examples in without assuming Mu(X)<infinity neither L
q space is subset of L p space nor L p space is subset of L q space.

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Clarification of Question by madukar-ga on 04 Feb 2003 12:10 PST

here !!f!!p means p norm of f.