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

?