- #1

AxiomOfChoice

- 533

- 1

[tex]

\|f + g + h\|^2 + \|f - h - g\|^2 = 2\|f - h\|^2 + 2\|g\|^2.

[/tex]

They attempt to justify this using the parallelogram law:

[tex]

\|x + y\|^2 + \|x - y\|^2 = 2\|x\|^2 + 2\|y\|^2,

[/tex]

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