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For an infinite Hamel basis, its dual family is linearly independent but does not span the algebraic dual

Statement

Let $B$ be an infinite Hamel basis of $V$ . Its dual family $(b^{*})_{b \in B}$ is linearly independent in $V^{*}$ but does not span $V^{*}$ .

Facts & Assumptions

Given: An infinite Hamel basis $B$ of $V$ and its dual family.

[L1]

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

[L2]

Linear independence tests only finite linear relations (Linear independence: a finite list $v : n \to V$ is independent when $\sum_{i < n} λ_{i} v_{i} = 0_{V}$ forces every $λ_{i} = 0_{F}$ , and a subset $S \subseteq V$ is independent when every injective finite list into $S$ is independent).

[L3]

A Hamel basis is a linearly independent spanning set, with span defined through finite linear combinations (Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis).

Proof

technique · direct

1.1

If a finite relation $\sum_{b \in S} a_{b} b^{*} = 0$ holds, evaluating it at each $c \in S$ gives $a_{c} = 0$ by [L1]. Thus the dual family is linearly independent by [L2].

L1L2

1.2

By [L3], every vector has a finite basis expansion. It is unique after zero coefficients are discarded, because subtracting two such expansions gives a finite relation in the independent set $B$ . Hence $ϕ (\sum_{b \in S} a_{b} b) = \sum_{b \in S} a_{b}$ is well defined and linear, and $ϕ (b) = 1$ for every $b \in B$ .

L3algebra

2.1

Every finite linear combination of members of the dual family vanishes at all basis vectors outside its finite support. Since $B$ is infinite while $ϕ (b) = 1$ for every $b$ , $ϕ$ is not in their span.

step 1.2L1given

3.1

The dual family is therefore independent but not spanning, so it is not a Hamel basis of $V^{*}$ .

step 1.1step 2.1∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 53 results over 18 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

K. Conrad, Infinite-Dimensional Dual Spaces, Theorem 1 (standard reference, not scraped)