Dual norms



(In the following, we may work on R^n, though the same questions arise
in other spaces.)

Consider two norms | |_a and | |_b. Then

phi:v -> (|v|_a^p + |v|_b^p)^{1/p}

is itself a norm. This is a special case of a much more general
statement - a norm of norms is a norm.

Question: what is the dual norm of phi?

For example, what is the dual norm of

v -> (c_1 |v|_1^2 + c_2 |v|_2^2)^{1/2},

where c_1,c_2 are positive numbers? (I do not know the answer even for
c_1=1, c_2=2.)

Harald
PS. The answer is not ((1/c_1) |v|_{\infty}^2 + (1/c_2) |v|
_2^2)^{1/2}, as one might think at first.
.