I have worked my way though the proof of the Cauchy Schwarz inequality in Rudin but I am struggling to understand how one could have arrived at that proof in the first place. The essence of the proof is that this sum:(adsbygoogle = window.adsbygoogle || []).push({});

##\sum |B a_j - C b_j|^2##

is shown to be equivalent to the following expression:

##B(AB - |C|^2)##

Now since each term of the first sum is positive, it is clearly greater than or equal to zero, so that the expression $$B(AB - |C|^2)$$ is also greater than or equal to zero. Now if $$B = 0$$ the theorem is trivial, so assume that $$B \geq 0$$ and then the inequality $$B(AB - |C|^2) \geq 0$$ implies that $$AB - |C|^2 \geq 0$$ which is the theorem.

Now naturally what I want to understand is how to arrive at this proof in the first place. Some intuition to start with is that if $$AB - |C|^2$$ can be made equivalent to a single sum, each term of which is nonnegative, this would give the desired result. But Rudin added a step to this, by showing that $$B(AB - |C|^2)$$ can be made equivalent to a single sum and then the B can be cancelled out. What train of thought would have led Rudin to this proof?

There is an explanation offered here:

http://math.berkeley.edu/~gbergman/ug.hndts/06x2+03F_104_q+a.txt

But I am still struggling to figure out that explanation too. Can anyone either help or direct me to a useful resource?

Thanks!

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Cauchy schwarz inequality in Rudin

Loading...

Similar Threads - Cauchy schwarz inequality | Date |
---|---|

I Cauchy's Theorem | May 22, 2017 |

Cauchy sequences or not | Dec 7, 2015 |

Why isnt Cauchy's formula used for the perimeter of ellipse? | Aug 17, 2015 |

Complex Analysis: Use of Cauchy | May 7, 2015 |

Schwarz inequality is Cauchy–Schwarz inequality? | Oct 30, 2012 |

**Physics Forums - The Fusion of Science and Community**