CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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The polynomial space admits no complete norm

Statement refuted

Refuted claim: the polynomial space over a scalar field can be made into a Banach space by some norm.

Let $K \in {R, C}$ and let $K [x]$ be the polynomial ring of The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution, regarded as a vector space over $K$ . Then no norm on $K [x]$ is complete.

Facts & Assumptions

Given: A scalar field $K \in {R, C}$ and the vector space $K [x]$ of polynomials in one indeterminate.

[L1]

A Banach space has no countably infinite Hamel basis (A Banach space has no countably infinite Hamel basis).

[L2]

A basis is a linearly independent spanning subset (Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis).

[L3]

The polynomial ring $K [x]$ is the set of finite sums $a_{0} + a_{1} x + \dots + a_{m} x^{m}$ (The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution).

Counterexample

technique · direct

1.1L2L3algebra

The set ${1, x, x^{2}, \dots}$ spans $K [x]$ by [L3], because every polynomial is a finite linear combination of monomials. It is linearly independent: if $a_{0} + a_{1} x + \dots + a_{m} x^{m} = 0$ as a polynomial, then every coefficient is $0$ . Hence [L2] makes ${1, x, x^{2}, \dots}$ a countably infinite Hamel basis of $K [x]$ .

2.1L1step 1.1assume-contradischarge-contradiction∎

If some norm on $K [x]$ were complete, then [L1] would forbid the countably infinite Hamel basis from step 1.1. This contradiction shows that no norm on $K [x]$ is complete.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

28 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Christopher Heil, A Basis Theory Primer (standard reference, not scraped)