On Problems 4 and 5 with 1<p< infinity and q the index dual to p, it will be
helpful to remember from the proof of Young's inequality
that, for y and z positive numbers,
y^p = z^q if and only if
y=z^(q-1) if and only if
z=y^(p-1) and, under these equivalent conditions,
zy = y^p =z^q.
See attached file for full problem description.
The Riesz Representation Theorem and Hoelder's Inequality are investigated.