How I prove the triangle ineqaulity by the inequalty of Cauchy-Swartz?
Thanks, I refer to:
https://en.wikipedia.org/wiki/Cauchy...arz_inequality
You understand that we do not have a copy of the notes/textbook from which you are working?
Here is a classic proof if one is working with absolute value as a norm.
I have absolutely no idea what any of that means.
Suppose that each of $a~\&~b$ is a real number.
$ \begin{align*}|a|\cdot |b|&=|a\cdot b| \\&\ge a \cdot b \end{align*}$ so that $2|a|\cdot |b|\ge 2a\cdot b$.
$ \begin{align*}|a+b|^2&=(a+b)^2 \\&=a^2+2a\cdot b+b^2\\&\le |a|^2+2|a\cdot b|+b^2\\ &\le a^2+2|a|\cdot
|b|+|b|^2\\&\le (|a|+|b|)^2 \end{align*}$
Using the square root we get the result.