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Inner products separate vectors, and the induced norm is homogeneous:
Statement
In a real or complex inner product space:
- if for every , then ;
- for every scalar and vector .
Facts & Assumptions
Given: Vectors in an inner product space and a scalar .
Positive definiteness says exactly when , and the inner product is linear first and conjugate-linear second (Real and complex inner product spaces, with the inner product linear in the first argument).
The induced norm is the unique nonnegative square root of (The norm induced by a real or complex inner product, Existence and uniqueness of -th roots: a unique with ).
Proof
If for every , take ; [L1] gives .
Sesquilinearity and [L3] give . Both and are nonnegative, so uniqueness in [L2] gives their equality.
Depends on
- Real and complex inner product spaces, with the inner product linear in the first argument
- The norm $\lVert v\rVert=\sqrt{\langle v,v\rangle}$ induced by a real or complex inner product
- Conjugation is an involutive real-field automorphism, $z\overline z=|z|^2$, and modulus is definite, multiplicative, and subadditive
- Existence and uniqueness of $n$-th roots: a unique $a^{1/n} \ge 0$ with $(a^{1/n})^n = a$
Used by
- The inner-product norm is definite, homogeneous, and satisfies the triangle inequality Corollary
- The quaternion double cover generates the third homotopy group of SO(3) Lemma
- Adjoints satisfy (S+T)^*=S^*+T^*, (λ T)^*=λ̄ T^*, (ST)^*=T^*S^*, and T^**=T Proposition
- Every linear map between finite-dimensional inner product spaces has a unique adjoint Theorem
- Finite-dimensional Riesz representation: every functional is uniquely v↦⟨ v,w⟩ Theorem
- For an endomorphism in finite dimension, preserving lengths, preserving inner products, carrying orthonormal bases to orthonormal bases, and T^*T=I are equivalent Theorem
Dependency tree · two levels
26 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Sheldon Axler, Linear Algebra Done Right, 4th ed., §6A (standard reference, not scraped)