When F = ℝ or ℂ, there is something called an inner product. In case of ℝ it is also called the dot product. This is also often referred to as the scalar product.
Definition 1.16.1 Let a,b ∈ F^{n} define a ⋅ b as

This will also be denoted as
With this definition, there are several important properties satisfied by the inner product. In the statement of these properties, α and β will denote scalars and a,b,c will denote vectors or in other words, points in F^{n}.
Proposition 1.16.2 The inner product satisfies the following properties.
 (1.19) 
 (1.20) 
 (1.21) 
 (1.22) 
 (1.23) 
You should verify these properties. Also be sure you understand that 1.22 follows from the first three and is therefore redundant. It is listed here for the sake of convenience.
Example 1.16.3 Find
This equals 0 + 2
The Cauchy Schwarz inequality takes the following form in terms of the inner product. I will prove it using only the above axioms for the inner product.
Theorem 1.16.4 The inner product satisfies the inequality
 (1.24) 
Furthermore equality is obtained if and only if one of a or b is a scalar multiple of the other.
Proof: First define θ ∈ ℂ such that

and define a function of t ∈ ℝ

Then by 1.20, f
f  = a ⋅ +
tθb ⋅ 
= a ⋅ a + tθ +
tθ +
t^{2} ^{2}
b ⋅ b 

Now if

since otherwise the function, f
It is clear from the axioms of the inner product that equality holds in 1.24 whenever one of the vectors is a scalar multiple of the other. It only remains to verify this is the only way equality can occur. If either vector equals zero, then equality is obtained in 1.24 so it can be assumed both vectors are non zero. Then if equality is achieved, it follows f
You should note that the entire argument was based only on the properties of the inner product listed in 1.19  1.23. This means that whenever something satisfies these properties, the Cauchy Schwarz inequality holds. There are many other instances of these properties besides vectors in F^{n}. Also note that 1.24 holds if 1.20 is simplified to a ⋅ a ≥ 0.
The Cauchy Schwarz inequality allows a proof of the triangle inequality for distances in F^{n} in much the same way as the triangle inequality for the absolute value.
Theorem 1.16.5 (Triangle inequality) For a,b ∈ F^{n}
 (1.25) 
and equality holds if and only if one of the vectors is a nonnegative scalar multiple of the other. Also
 (1.26) 
Proof: By properties of the inner product and the Cauchy Schwarz inequality,



Taking square roots of both sides you obtain 1.25.
It remains to consider when equality occurs. If either vector equals zero, then that vector equals zero times the other vector and the claim about when equality occurs is verified. Therefore, it can be assumed both vectors are nonzero. To get equality in the second inequality above, Theorem 1.16.4 implies one of the vectors must be a multiple of the other. Say b = αa. Also, to get equality in the first inequality,

Therefore, α must be a real number which is nonnegative.
To get the other form of the triangle inequality,

so

Therefore,
 (1.27) 
Similarly,
 (1.28) 
It follows from 1.27 and 1.28 that 1.26 holds. This is because