There's a theorem in my Linear Algebra book which states:
Let
and
be two vectors in an inner product space
, such that
. Then
Here
means the distance between two vectors.
The proof is given like this:
Let
. Then you write:
where
and
are orthogonal. You can verify this by using the inner product
axioms to show that
....rest of the proof.
And Inner product axioms are:
Let
and
be vectors in a vector space
, let
be any scalar. An inner product on
is a function that associates a real number
with each pair of vectors
and
and satisfies the following axioms.
1)
2)
3)
4)
My question: How do you prove:
using inner product axioms? I can't make a connection with the inner product axioms and this.
As you seen above the author is saying that this can be proved using inner product axioms. Is it possible to kindly show that
?