Alphabeta Math
DefinitionDefinition: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck pass
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Finite complex-array norms used by the NUFFT certificates

Definition

For a finite index set E and a complex array z=(ze)e∈E, define ∥z∥1=∑e∈E∣ze∣,∥z∥2=(∑e∈E∣ze∣2)1/2,∥z∥∞=max⁡e∈E∣ze∣. For z,w∈CE, define the counting inner product by ⟨z,w⟩E=∑e∈Ezewe‾. By Conjugation is an involutive real-field automorphism, zz‾=∣z∣2, and modulus is definite, multiplicative, and subadditive, ⟨z,z⟩E=∑e∈E∣ze∣2, so ∥z∥2=⟨z,z⟩E. For E=∅, all three norms and this inner product are defined to be zero. For nonempty finite E, the maximum exists by repeated comparison in the real order. Complex modulus is as in Real intervals, complex rectangles and ball enclosures, conjugation as in Real and imaginary parts, complex conjugation, and modulus, finite sums as in A finite sum in a commutative monoid indexed by an arbitrary finite set, and the square root is the unique nonnegative root.

Facts & Assumptions

Given: A finite set E and a complex array z∈CE.

[F1]

A finite sum in a commutative monoid indexed by an arbitrary finite set: a finite indexed sum is formed from finitely many terms and the empty sum is zero.

[F2]

Real intervals, complex rectangles and ball enclosures: complex modulus is a finite nonnegative real number.

[F3]

Nonnegative real square roots exist uniquely: every nonnegative real has a unique nonnegative square root.

[F4]

Dedekind-complete ordered real field: the real order is total, so any two real numbers are comparable.

[F5]

The principle of mathematical induction: a statement proved at the first finite list length and preserved when one entry is added holds for every finite list length.

Proof

1.1givenF1F2F4F5induction

For E=∅ all three values are stipulated as zero. If E is nonempty finite, enumerate its elements and induct on the number of entries. A one-entry list of real moduli has a maximum; given a maximum for the first n entries, compare it with the (n+1)st modulus using [F2] and [F4] and keep the larger. Induction [F5] constructs the maximum of the finite modulus list. The finite sum formulas in [F1] are well-defined.

2.1F1F2F3step 1.1algebra∎

Every term ∣ze∣2 is nonnegative by [F2], so the finite sum in the Euclidean norm is nonnegative; [F3] supplies its unique nonnegative square root. Thus all three displayed values are defined, including the separately stipulated empty-array case.

Depends on

Used by

Dependency tree · two levels

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Sources