Parallelogram law calculation is this an error in the text?

[AxiomOfChoice]

In Kolmogorov and Fomin's Real Analysis book, pg. 161, they make the following claim: For any vectors f,g,h in a real Hilbert space, we have

<br /> \|f + g + h\|^2 + \|f - h - g\|^2 = 2\|f - h\|^2 + 2\|g\|^2.<br />

They attempt to justify this using the parallelogram law:

<br /> \|x + y\|^2 + \|x - y\|^2 = 2\|x\|^2 + 2\|y\|^2,<br />

which holds in any inner product space. But I do not think they're right about this; doesn't their claim fail in \mathbb R with f = 2, g = -1, h = 1, when the inner product is just multiplication? Don't you get something like 8 = 4?

 

[LCKurtz]

If the first h in your equation had a negative in front of it you would get the result using x=f-h and y=g. Maybe it is just a typo.

 

[AxiomOfChoice]

For anyone who should happen across this page in the future: I've discovered that this actually is a typo. Click http://math.gmu.edu/~tlim/errataByEdgar.pdf" for more information. Apparently, that entire section of the book (examining when a norm is derived from an inner product) is littered with errors.