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For the polynomial space $F [x]$ , the canonical map to the algebraic double dual is injective but not surjective

Example

Assume the axiom of choice. For the polynomial vector space $V = F [x]$ , the canonical map $J_{V} : V \to V^{* *}$ is injective but not surjective.

Facts & Assumptions

Given: The axiom of choice and a field $F$ .

[L1]

A polynomial over $F$ is a coefficient sequence with finite support (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution).

[L2]

Under Choice, the canonical map is onto exactly in finite dimension and is always injective (Assuming choice, $J_{V} : V \to V^{* *}$ is surjective if and only if $V$ is finite-dimensional).

[L3]

The coordinate functionals of an infinite Hamel basis span a proper subspace of the dual (For an infinite Hamel basis, its dual family is linearly independent but does not span the algebraic dual), and a vector outside a subspace can be separated from it by a functional (Assuming choice, if $v \notin U \leq V$ , some $f \in V^{*}$ vanishes on $U$ and satisfies $f (v) = 1$ ).

Verification

technique · explicit double-dual witness

1.1

By [L1], the monomials $1, x, x^{2}, \dots$ form an infinite Hamel basis. Let $δ_{n}$ extract the coefficient of $x^{n}$ , and put $Φ = span {δ_{n} : n \geq 0}$ . Define $ϕ (\sum_{n} a_{n} x^{n}) = \sum_{n} a_{n}$ ; finite support makes this a functional. Every member of $Φ$ vanishes on all but finitely many monomials, whereas $ϕ (x^{n}) = 1$ for every $n$ , so $ϕ \notin Φ$ .

L1L3algebra

2.1

Apply the separation statement in [L3] inside $V^{*}$ to choose $L \in V^{* *}$ with $L ∣_{Φ} = 0$ and $L (ϕ) = 1$ . If $L = J_{V} (p)$ , then $0 = L (δ_{n}) = δ_{n} (p)$ for every $n$ , so all coefficients of $p$ vanish and $p = 0$ ; this would give $L = 0$ , contradicting $L (ϕ) = 1$ .

step 1.1L3choose

3.1

Thus $L$ is outside the image, while injectivity follows from [L2]. This explicitly realizes the finite-dimensional boundary in [L2].

step 2.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 48 results over 14 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 (standard reference, not scraped)

For the polynomial space F[x]F[x], the canonical map to the algebraic double dual is injective but not surjective