I've been reading "The Qualitative Theory of Ordinary Differential Equations, An Introduction" and am now stuck on an inequality I am supposed to be able to prove. I am pretty sure the inequality comes from linear algebra, I remember seeing something about it in my intro class but I let a friend borrow my book. I have been unable to find an explanation of the 2|uv| inequality, which may be all I need.(adsbygoogle = window.adsbygoogle || []).push({});

Definitions where bold indicates vectors.

[tex]\text{Euclidean Length: }||\textbf{y}|| = \left(\Sigma_{i=1}^n |y_i|^2\right)^{1/2}, \text{Norm: } |\textbf{y}| = \Sigma_{i=1}^n|y_i| [/tex]

[tex]

\text{To Prove:} \\

\text{If } \textbf{y} \in E_n, \text{ show that} \\

||\textbf{y}|| \leq |\textbf{y}| \leq \sqrt{n}||\textbf{y}|| \\

\text{Hint, use } 2|uv| \leq |u|^2 + |v|^2 \text{ and show that } ||\textbf{y}||^2 \leq |\textbf{y}|^2 \leq n||\textbf{y}||^2[/tex]

I was thinking that generalizing the [itex]2|uv|[/itex] inequality to

[tex]n|u_1 u_2 \cdots u_n| \leq |u_1|^2 + |u_2|^2 + \cdots + |u_n|^2[/tex]

may lead me in a helpful direction, but couldn't see where to go with it, or where that inequality came from to generalize it.

That may not even be the right way to be thinking about the proof, but any direction you could give me in this direction would be greatly appreciated. Thank you!

-Sravoff

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# Length-Norm inequality to root-n times Length (using 2|uv| <= |u|^2 + |v|^2)

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