Theorems · Theorem · functional analysis

NormedSpace.polar_ball_subset_closedBall_div

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : SeminormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {c : 𝕜},
  1 < ‖c‖ → ∀ {r : ℝ}, 0 < r → StrongDual.polar 𝕜 (Metric.ball 0 r) ⊆ Metric.closedBall 0 (‖c‖ / r)

Defined in: Mathlib.Analysis.Normed.Module.Dual
Cited by: 1 results in Mathlib
Foundations: Depth 172 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NontriviallyNormedField SeminormedAddCommGroup NormedSpace

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Cites20

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coeproof · cited by 62,936
Setstatement · cited by 53,352
Realstatement and proof · cited by 25,697
RingHom.idstatement · cited by 18,349
NormedSpacestatement and proof · cited by 12,499
NontriviallyNormedFieldstatement and proof · cited by 8,742
Norm.normstatement and proof · cited by 5,413
SeminormedAddCommGroupstatement and proof · cited by 2,671
LT.lt.leproof · cited by 2,189
Metric.ballstatement and proof · cited by 735
Metric.closedBallstatement · cited by 704
zero_lt_oneproof · cited by 598

Cited by1

Results whose statement or proof uses this declaration.

NormedSpace.isBounded_polar_of_mem_nhds_zeroproof · cited by 1