User talk:Arjunarikeri

Hello.

Your edit to Cauchy–Schwarz inequality seems to assume that


 * $$\langle x, y \rangle + \langle y, x \rangle = 2|\langle x, y \rangle|. $$

That is true only if $$\langle x, y \rangle$$ is a positive real number. It is not true if $$\langle x, y \rangle$$ is negative, and it is not true if $$\langle x, y \rangle$$ is a complex number with a non-zero imaginary part. In this last case, one must remember that $$\langle x, y \rangle$$ and $$\langle y, x \rangle$$ (with y and x in the opposite order) are not equal to each other, but are complex conjugates of each other. For example, if


 * $$ \langle x, y \rangle = 3 + 4i\,$$

then


 * $$ \langle y, x \rangle = 3 - 4i\,$$

("+" has changed to "&minus;"), and then


 * $$\langle x, y \rangle + \langle y, x \rangle = (3+4i) + (3 - 4i) = 6\,$$

whereas


 * $$2|\langle x, y \rangle| = 2 |3+4i| = 2\sqrt{3^2 + 4^2\, } = 2 \cdot 5 = 10.$$

That is why I reverted. Michael Hardy 21:37, 23 August 2006 (UTC)