Re: Dual norms



On Apr 16, 2:45 am, Harald Helfgott <harald.helfg...@xxxxxxxxx> wrote:
(In the following, we may work on R^n, though the same questions arise
in other spaces.)

Taking the questions in reverse order:

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.

Let us specialize to R^2, as I think you have. Then the norm of (x,y)
is
sqrt(c_1*x^2 + c_2*y^2). The dual norm is the greatest norm of a unit
vector, which would be sqrt(max(c_1,c_2))

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?
I don't believe there is a formula in term of the dual norms of | |_a
and | |_b. The norms na=sqrt(x^2+y^2), nb=ncsqrt(x^2+y^2/100), and
nc=sqrt(x^2/100+y^2) all have dual norm 1, but the dual norm of (na^p
+na^p)^(1/p) is 2^(1/p) while the dual norm of (nb^p+nc^p)^(1/p) is
lower
.

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