Let be real-valued integrable functions on a measure space ,and define:
by Fubini theorem(cake representation theorem in fact):
(there is a geometric heuristic,strict proof is due to fubini theorem)
Let be a infinite dimensional real Hilbert space.
a)Prove the unit sphere of is weakly dense in the unit ball of .
b)Prove there is a sequence of bounded linear operator from to such that for all n but for all .
by Zorn lemma there is a orthogonal bases .
to proof a),suffice to proof:.
this can be done by look at the expansion .due to the Cauchy inequality,there is a freedom of choice the coefficient for .the choice will lead a).
b) is trivial due to a).
3.Let be a Banach space and let $X^*$ be it dual Banach space.Prove that if is separable then is separable.
we know is the space consist with bounded(continued) linear functional on .
for ,,so due to is separable.there is a countable dense set in .i.e. .we equip a member of to by ,.
On the other hand, we construct a functional ..so it is obviously to show is dense in .
5.Let be the unit interval,and for and ,let:
For define .
Prove that if then a.e. in I.
10.Let be the open unit disc and be Lebesgue measure on .let be the subspace of consisting of holomorphic functions.Show that is complete.
proof:maximum norm principle.to show is closed uniformly coverage.so is harmonic and is due to itself is complete.