- #1
babyrudin
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Homework Statement
For [tex]x,y \in R^n[/tex], their inner ("dot") product is given by
[tex]<x,y>=\sum_{i=1}^n x_i y_i.[/tex]
Also, we write
[tex]<x,x>=\|x\|^2.[/tex]
Homework Equations
Fix p>1. Show that for all [tex]x,y \in R^n[/tex] we have
[tex]< \|x\|^{p-2}x -\|y\|^{p-2}y, x-y> \geq 0[/tex]
The Attempt at a Solution
Expanding the left-hand side, we can write
[tex]<\|x\|^{p-2}x,x> -<\|y\|^{p-2}y,x>-<\|x\|^{p-2}x,y>+<\|y\|^{p-2}y,y>[/tex]
which further simplifies to
[tex]\|x\|^p +\|y\|^p -(\|x\|^{p-2} +\|y\|^{p-2})<x,y>.[/tex]
Then I'm stuck. How do I show that the above is nonnegative?
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