Lemma (mathematics)

Characteristics

Lemmas are typically narrower in scope than theorems: they address a specific technical point needed inside a larger proof. They may be routine or intricate, short or elaborate, but their role is functional rather than celebratory. Authors use lemmas to break proofs into manageable pieces, to avoid repetition, and to clarify dependencies within an argument.

Origin and historical note

The word derives from the Greek λεμμα (lemma), meaning "something taken" or "assumption." The practice of naming intermediate results has a long history in mathematical writing, from classical geometry through modern analysis. For more on etymology and historical usage see background material.

Uses and notable examples

• Euclid's lemma: a basic fact about prime divisors used in elementary number theory.
• Zorn's Lemma: a key principle in set theory and algebra, equivalent to the axiom of choice.
• Fatou's Lemma: an inequality used in measure theory and integration.

These illustrate how lemmas can range from elementary facts to deep statements that are central across fields. For suggestions on applying lemmas in proofs see practical guidance.

Distinctions and notable facts

• Lemma vs theorem: a label of convenience; sometimes a famous lemma is more influential than many theorems.
• Corollary: a direct consequence of a theorem or lemma, usually immediate.
• Named lemmas often become standard tools and acquire importance beyond their original context.

Writers choose the term "lemma" to emphasize function. Whether a result is called a lemma, proposition, or theorem depends on tradition, emphasis, and style rather than a strict hierarchy. See further reading here.