Cauchy-Schwarz Inequality
The Cauchy-Schwarz inequality is the upper bound of the inner product.
Contents
Description
The inequality is defined as |⟨u, v⟩|2 ≤ ⟨u, u⟩⟨v, v⟩ for two non-zero vectors u and v.
Note that the natural norm for an inner product space is defined as ||a|| = √⟨a, a⟩. By taking the square root of both sides of the Cauchy-Schwarz inequality, the triangle inequality is defined.
The inequality is sometimes proven in the complex plane. Besides noting that C is isomorphic to R2, Hilbert spaces can be real or complex, so it is necessary to always consider the complex case.
Proof
Note that the projection of u onto v is calculated as 
. Note furthermore that a right triangle is formed by u, the projection of u onto v, and the orthogonal component that can be calculated as 
.
Since the original vectors are non-zero, the norm of this orthogonal component is characterized by:
In the case that the inner product space is real, this expands to:
This can be simplified as follows:
This can now be easily rewritten into the familiar form of the Cauchy-Schwarz inequality.
Consider now the case of a complex inner product space. The inner product must be Hermitian, so order matters; 
. Furthermore, the expansion requires a complex conjugate. The norm now evaluates as:
This can be simplified as follows:
Leveraging now the definition that ⟨u, v⟩ is the conjugate of ⟨v, u⟩:
Once again this can be easily rewritten into the familiar form of the Cauchy-Schwarz inequality.
