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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.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 50 results over 17 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

H. Pinkham, Linear Algebra, §6.1 (standard reference, not scraped)
K. Conrad, Infinite-Dimensional Dual Spaces (standard reference, not scraped)

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