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| Note that the [[LinearAlgebra/Norm|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. | 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. |
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| The inequality is sometimes proven in the complex plane. This can be a simpler proof and, since ''C'' is isomorphic to ''R^2^'', it is entirely equivalent. | 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. ---- == 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}} |
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.
