### Cauchy-Schwarz Inequality

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

with equality if and only if for some scalar . 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

or, removing the normalization,

The same derivation holds if is replaced by yielding

The last two equations imply

In the complex case, let , and define . Then is real and equal to . By the same derivation as above,

Since , the result is established also in the complex case.
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