Theorem

Weierstrass Approximation Theorem

Analysis

Every continuous function on a closed, bounded interval of the real line can be uniformly approximated as closely as desired by a polynomial. Proved by Karl Weierstrass, it underlies much of approximation theory and numerical analysis.

Facts

Statement

Every continuous real-valued function on a closed real interval [a, b] can be uniformly approximated as closely as desired by a polynomial function. 1

Proof Year

1885 1

Classification

Statement Form

Existence Theorem 1

Connections

Has Statement Form

Existence Theorem

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

In Branch

Analysis

Proved By

Karl Weierstrass

Sources

1. Weierstrass Approximation Theorem (Wikipedia, Stone-Weierstrass Article)

Wikimedia Foundation

Stone-Weierstrass theorem article, statement box opening clause
Suppose f is a continuous real-valued function defined on the real interval [a, b].
History note on Weierstrass's 1885 proof
The original version of this result was established by Karl Weierstrass in 1885 using the Weierstrass transform.

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Weierstrass Approximation Theorem

Has Statement Form

In Branch

Proved By