Cauchy-Schwarz Inequality
The Cauchy-Schwarz inequality is an inequality relating the inner products of vectors in some inner product space.
For all
Written in terms of the norm induced by the inner product the Cauchy-Schwarz inequality is:
It is one of the most fundamental inequalities useful to prove many other inequalities in real analysis.
This is very easy to see in the case of the real dot product where it follows from the fact that
Proof
Let
Then consider:
This is almost the desired inequality, however the absolute values are missing on the right hand side. Since
And then finally, we can square both sides since they are both positive:
For Real Coordinate Space
Given
This is a special case of Holder's inequality.