About Inner Product Spaces?

Question

If In An Inner Product Space = for all x, show that u = v

Answer 1

Suppose ∀x, = . Then in particular, = , so - = 0. But by linearity of the inner product, = - , so this means = 0, so u-v = 0, so u=v. Q.E.D.

Answer 2

No. the indoors product is a function, usually represented via a dot . defined in V x V and with values in R, such that, for each x, y and z in V: x.x >=0 with equality if and in basic terms x =0 x. y = y. x x.(y+z) = x.y + x.z and (x + y) . z = x.y + x.z (a x).y = a x.y = x.(ay) we are saying x and y are orthogonal if x.y = 0. So, the orthogonality of x and y relies upon on the form you defined the indoors product . in accordance to such definition x and y could or won't be orthogonal. Orthogonality relies upon on the set and on the indoors product defined on it, no longer in basic terms on the set.