Alphabeta Math

R^n as a Normed Space; Vector-Valued Functions

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✓ 13 results · all verified · 4 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full by a delegated reviewing agent on the owner's instruction; the judge is an additional, independent cross-model AI review of the proofs. The 9 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Linear Algebra
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Definition

A norm on a real vector space, the induced metric, and the dictionary with the metric axioms

Definition

The Euclidean inner product ⟨ x,y⟩ = ∑_k<n xₖ yₖ on ℝⁿ

Theorem

Cauchy-Schwarz |⟨ x,y⟩| ≤ ‖ x‖₂‖ y‖₂ with its equality case, the triangle inequality for ‖·‖₂, the parallelogram law and polarisation

Definition

The p-norms ‖ x‖ₚ for rational p ≥ 1, and ‖ x‖_∞

Lemma

Each ‖·‖ₚ is a norm on ℝⁿ, and the induced metrics are exactly d₁, d₂ and d_∞ of the published metric-spaces page

Definition

Equivalent norms, and the dictionary with equivalent metrics

Lemma

The finite and reverse triangle inequalities for a norm; and for n ≥ 1 every norm N on ℝⁿ satisfies N(x) ≤ C‖ x‖₁ and is Lipschitz, hence continuous, for d₂

Theorem

For n ≥ 1 all norms on ℝⁿ are equivalent

Theorem

For n ≥ 1 a sequence in ℝⁿ converges iff each coordinate sequence converges, is Cauchy iff each coordinate sequence is Cauchy, and ℝⁿ is complete in every norm

Corollary

For n ≥ 1 every bounded sequence in ℝⁿ has a convergent subsequence

Definition

Vector-valued functions f : A → ℝᵐ, their limits and continuity, with the dictionary to the metric notions

Theorem

A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions

Definition

The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral

Theorem

For a ≤ b and f : [a,b] → ℝᵐ integrable when a<b, ‖∫ₐᵇ f‖₂ ≤ ∫ₐᵇ ‖ f‖₂; for a<b, ‖ f‖₂ is integrable

Theorem

The mean value inequality: if f : [a,b] → ℝᵐ is continuous and differentiable on (a,b) with ‖ f'‖₂ ≤ M, then ‖ f(b)-f(a)‖₂ ≤ M(b-a)

Corollary

If f : [a,b] → ℝᵐ is differentiable with integrable f' then ∫ₐᵇ f' = f(b)-f(a); and a bounded derivative makes f Lipschitz

Definition

Series of vectors in ℝⁿ, absolute convergence, rearrangement, and the set of rearrangement sums

Theorem

An absolutely convergent series in ℝⁿ converges, and every rearrangement converges to the same sum

Definition

The subspace Γ of directions along which a series converges absolutely, and its orthogonal complement Γ^⊥

Theorem

Steinitz's polygonal confinement theorem: finitely many vectors of norm at most 1 summing to 0 can be ordered so that every partial sum has norm at most n

Theorem

The set of rearrangement sums of a convergent series in ℝⁿ is a nonempty subset of the affine subspace s + Γ^⊥

Remark

Conventions of this page, the standing n ≥ 1 hypothesis, and what is taken up elsewhere in the reading order

Sources