A seminorm with two-sided bounds against a reference norm

MathlibAnnex.EquivalentSeminorm

structure

Records a norm through a seminorm, two positive comparison constants and continuity on the original normed space.

Statement

Let be a real normed space with reference norm . An equivalent-seminorm datum consists of a real seminorm and constants such that

together with continuity of for the reference topology. The record keeps this comparison data explicit.

Definition

The eight fields are: (1) the seminorm ; (2) a real lower constant ; (3) a real upper constant ; (4) a proof that ; (5) a proof that ; (6) the lower comparison for every ; (7) the upper comparison for every ; and (8) continuity of . These fields specify data and its properties; constructing such a record requires supplying all eight.

Assumptions

The carrier is a real normed vector space. The stored function is a seminorm: it is subadditive, nonnegative and satisfies . Both comparison constants are strictly positive, both inequalities hold for every , and continuity of is part of the stored data. There is no finite-dimensionality or completeness assumption.

Conclusion

The lower bound makes definite: if , then ; since , and . This is the separately proved definiteness lemma. The related definitions give and . In particular , by the bounded-ball lemma.

The reference norm on remains in place. The separate construction equips a copy of the carrier with norm ; its distance is , as recorded in the cited transport theorem. Boundedness of alone does not assert compactness in an arbitrary infinite-dimensional space.

structure EquivalentSeminorm (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E] where
  p : Seminorm ℝ E
  lower : ℝ
  upper : ℝ
  lower_pos : 0 < lower
  upper_pos : 0 < upper
  lower_le : ∀ x, lower * ‖x‖ ≤ p x
  le_upper : ∀ x, p x ≤ upper * ‖x‖
  continuous_p : Continuous p

namespace MathlibAnnex

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