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## Homework Statement

Let X = ##\mathbb{R^m}## and ||.|| be a Norm on X. The dual norm is defined as ##||y||_*:=sup({\langle\,x,y\rangle :||x|| \leq 1})##

a) Show that ##||.||_*## is also a norm

b) Construct two norms ##||.||^O## and ##||.||^C## so that:

{##x:||x||^O=1##} is a regular octahedron

and

{##x:||x||^C=1##} is cube<

I have a problem with b)

## Homework Equations

Definition of Norm

## The Attempt at a Solution

Now I've read that the One-Norm defines a Octahedron and the dual Norm a cube.

So {##x:||x||^O:=||x||_1 = |x|+|y|+|z| = 1##}

Now I have a problem to construct the dual norm since I don't fully understand dual norms.

But from the definition we get the cube ##||x||_*=sup(\langle\,x,y\rangle :||x||_1 \leq 1)##

But how do I do this since I have 3 entries and this inner product doesn't seem to fit?