Inner Product Space

Hey, Vilemiasma, glad to see that we've got a new contributor to the mathematics section! You were half-right about sesquilinarity. From the definitions above, that line was superfluous. However, you had gotten a bit confused in the linearity stuff. For a two-argument function like the inner product to be linear, it would have to act like this:

$\forall b\in F,\ \forall x,y\in V,\ \langle x,by\rangle =b\langle x,y\rangle$
$\forall b\in F,\ \forall x,y\in V,\ \langle bx,y\rangle =b\langle x,y\rangle$
$\forall x,y,z\in V,\ \langle x,y+z\rangle =\langle x,y\rangle +\langle x,z\rangle .$
$\forall x,y,z\in V,\ \langle x+y,z\rangle =\langle x,z\rangle +\langle y,z\rangle .$

Note that 2 is violated by the inner product, since we have:

$\forall b\in F,\ \forall x,y\in V,\ \langle bx,y\rangle ={\overline {b}}\langle x,y\rangle$

Also note that 3 & 4 are a part of the definition of linearity (and sesquilinearity), not a separate property.

I'm not sure if expanding sesquilinearity from one line into its components is a good idea, but I'll leave it that way for now.

Again, nice to have you on board. If you like, go check out the tips that Angela left at my talk page, it may help you get started with Wikipedia. -FunnyMan 18:21, Oct 23, 2004 (UTC)