DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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.

The dual family $(b^{})_{b \in B}$ associated to a Hamel basis $B$ , defined by $b^{} (c) = δ_{b c}$

Definition

Let $B$ be a Hamel basis of a vector space $V$ . For each $b \in B$ , the coordinate functional $b^{*} : V \to F$ is defined by

$b^{*} (c) = δ_{b c} (c \in B),$

and extended linearly: if the unique finite basis expansion of $v$ is $v = \sum_{c \in S} a_{c} c$ , then $b^{*} (v) = a_{b}$ , with $a_{b} : = 0$ when $b \notin S$ . Uniqueness of finite basis expansions makes this single-valued, and coordinatewise addition and scalar multiplication make $b^{*}$ linear. The family $(b^{*})_{b \in B}$ is the dual family associated to $B$ .

When $B$ is finite this family is the usual dual basis. When $B$ is infinite it remains a family in $V^{*}$ , but it need not span the whole algebraic dual.

The dual family $(b^{})_{b \in B}$ associated to a Hamel basis $B$ , defined by $b^{} (c) = δ_{b c}$

Definition

Depends on

Dependency tree · 2 levels

Sources

The dual family (b∗)b∈B(b∗)b∈B​ associated to a Hamel basis BB, defined by b∗(c)=δbcb∗(c)=δbc​

Definition

Depends on

Dependency tree · 2 levels

Sources

The dual family $(b^{})_{b \in B}$ associated to a Hamel basis $B$ , defined by $b^{} (c) = δ_{b c}$