Question

Let V be a vector space, with real numbers as scalars, which is equipped with
an inner product (
,
).
(a) Establish the identity
||x + y|| = ||x − y||,
where x and y are two orthogonal vectors in V

Answer 1

It is sufficient to show that the normed squared (ie both sides squared) are equal.
||x +y||^2=(x+y, x+y) the inner product
=(x,x) + (x,y) + (y, x) + (y, y)
but since x and y orthogonal
(x,y)=(y,x)=0
=> = (x,x)+(y,y) = ||x||^2 + ||y||^2
Now the RHS is the same process
||x-y||^2=(x-y, x-y) = (x,x) - (x, y) - (y, x) + (y, y)
same orthogonality argument
=||x||^2+||y||^2 so both sides are equal
||x + y||^2 = ||x-y||^2 implying the identity

Answer 2

Thanks so much appreciate it!! :)

Answer 3

A geometric way of thinking about this is that by virtue of the triangle inequality
|x+y| <= |x| + |y|, the distance function |x-y| = |y-x| is a metric for V.