Question 1 : a) 3 b) 3 c) 4

Question 2 : 10

Question 3 : a) 3 b) 3 c) 3 d) 1

Question 4 : a) 4 b) 3 c) 3

Question 5 : 10
Question 6 : 10
Question 7 : 6) 4)

Question 8 : a) 2 b) 1 c) 1 d) 4 e) 2


Question 1

Some students used that for any f in L^p there is g in L^p such that f=g a.e. and ess sup(f) = sup(g). HOWEVER this g need not be continuous, so one cannot say "f=g a.e. and are continuous so f=g".

Question 2

||f||=1 does not mean f is an isometry

Question 4
One needs to show that the function is well-defined, i.e. p(x) < \infty, that's where the condition 0 being in the interior is crucial.

Some examples for b)c) did not have 0 in the interior (like C = x-axis)

Question 5
Just a remark that euclidean was not necessary since there is only one Banach space of a fixed dimension up to isomorphism (all norms are equivalent), everything is isomorphic to L^2 which is self-dual.

Question 7
Some people got confused, and assumed that the maps Phi are surjective onto Y1 and Y2.