### Cauchy-Schwarz Inequality

The *Cauchy-Schwarz Inequality* (or ``Schwarz Inequality'')
states that for all
and
, we have

We can quickly show this for real vectors , , as follows: If either or is zero, the inequality holds (as equality). Assuming both are nonzero, let's scale them to unit-length by defining the normalized vectors , , which are unit-length vectors lying on the ``unit ball'' in (a hypersphere of radius ). We have

which implies

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