Sunday, 11 December 2016

Proving an inequality using Cauchy-Schwarz



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I am supposed to use the Cauchy Schwarz inequality to solve this problem but I am stuck. I can't see how the square root of N comes out. Could anyone please help me?


Answer



If x1,,xN are a basis of orthonormal vectors and ψ has a unit norm, ψ can be decomposed as
ψ=Nk=1αkxk,nk=1α2k=1
and
Nk=1

where by the Cauchy-Schwarz inequality
\left|\sum_{k=1}^{N}\alpha_k\right|^2\leq \sum_{k=1}^{N}1\sum_{k=1}^{N}\alpha_k^2 = N,
hence:
\sum_{k=1}^{N}\|\psi-x_k\|^2 \geq 2N-2\sqrt{N}
as wanted.


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