Alphabeta MathHow statement and proof provenance work
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The norm induced by a real or complex inner product
Definition
For a vector in a real or complex inner product space, define its inner-product norm by
Positive definiteness in Real and complex inner product spaces, with the inner product linear in the first argument makes the radicand a nonnegative real, and Existence and uniqueness of -th roots: a unique with supplies its unique nonnegative square root. The notation therefore defines one real number . Its norm axioms are established in The inner-product norm is definite, homogeneous, and satisfies the triangle inequality.
Depends on
Used by
- The inner-product norm is definite, homogeneous, and satisfies the triangle inequality Corollary
- Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces Definition
- Orthogonal vectors and subspaces, orthogonal and orthonormal sets, and orthonormal bases Definition
- Positive coordinate weights define an inner product and change lengths and projections Example
- Inner products separate vectors, and the induced norm is homogeneous: ‖λ v‖=|λ|‖ v‖ Lemma
- Pythagoras, the parallelogram identity, and the real and complex polarisation identities Proposition
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors Theorem
- Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 44 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Sheldon Axler, Linear Algebra Done Right, 4th ed., §6A (standard reference, not scraped)
- Sergei Treil, Linear Algebra Done Wrong, Ch. 5, §5.1 (standard reference, not scraped)