Inner product spaces and orthogonality?

Question

Let V be an inner product space. Let vectors u,v in V be such that u,v are orthogonal to each other. My question is that are we guaranteed that u,v will also be orthogonal under any other inner product defined?

In any vector space, it is possible to have more than one inner products defined. So if two vectors are orthogonal under one inner product, will they also be orthogonal under another inner product?

Answer 1

Let V=R^2 and define two inner products <,> and {,}as follows:
<(a,b),(c,d)>=ac+bd
{(a,b),(c,d)}=ac+2bd.
Then, (1,1) and (1,-1) are orthogonal in <,> but not in {,}.

Answer 2

No. The inner product is a function, usually represented by a dot . defined in V x V and with values in R, such that, for every x, y and z in V:

x.x >=0 with equality if and only 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 say x and y are orthogonal if x.y = 0. So, the orthogonality of x and y depends on how you defined the inner product . According to such definition x and y may or may not be orthogonal.

Orthogonality depends on the set and on the inner product defined on it, not just on the set.

Answer 3

No, u,v may be orthogonal in one inner product space and not orthogonal in another.

Answer 4

An inner product is linear and symmetric. that's <2x²+x,x> = 2 + = 2 + . So for this reason <2x²+x,x> = 2(-2) + 4 = 0. curiously, the 1st 2 words are already orthogonal.