= Cauchy-Schwarz Inequality =

The '''Cauchy-Schwarz inequality''' is the upper bound of the [[LinearAlgebra/InnerProduct|inner product]].

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== Description ==

The inequality is defined as ''|⟨u, v⟩|^2^ ≤ ⟨u, u⟩⟨v, v⟩'' for two non-zero vectors ''u'' and ''v''.

Note that the [[LinearAlgebra/Norm|natural norm]] for an [[LinearAlgebra/InnerProduct|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 ''R^2^'', [[Analysis/HilbertSpace|Hilbert spaces]] can be real or complex, so it is necessary to always consider the complex case.

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== Proof ==

Note that the [[Calculus/Projection#Vector_Projection|projection]] of ''u'' onto ''v'' is calculated as {{attachment:proj.svg}}. 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 {{attachment:orth.svg}}.

Since the original vectors are non-zero, the norm of this orthogonal component is characterized by:

{{attachment:norm.svg}}

In the case that the inner product space is real, this expands to:

{{attachment:real1.svg}}

This can be simplified as follows:

{{attachment:real2.svg}}

{{attachment:real3.svg}}

This can now be easily rewritten into the familiar form of the Cauchy-Schwarz inequality.

{{attachment:real4.svg}}

{{attachment:real5.svg}}

Consider now the case of a complex inner product space. The inner product must be [[LinearAlgebra/InnerProduct#Hermitian_Inner_Product|Hermitian]], so order matters; {{attachment:symmetry.svg}}. Furthermore, the expansion requires a [[Calculus/ComplexVector|complex conjugate]]. The norm now evaluates as:

{{attachment:complex1.svg}}

This can be simplified as follows:

{{attachment:complex2.svg}}

{{attachment:complex3.svg}}

Leveraging now the definition that ''⟨u, v⟩'' is the conjugate of ''⟨v, u⟩'':

{{attachment:complex4.svg}}

Once again this can be easily rewritten into the familiar form of the Cauchy-Schwarz inequality.

{{attachment:complex5.svg}}

{{attachment:complex6.svg}}



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