Theorems · Theorem · functional analysis
Seminorm.continuous_iff
∀ {𝕜 : Type u_3} {E : Type u_7} [inst : NontriviallyNormedField 𝕜] [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E]
[inst_3 : TopologicalSpace E] [IsTopologicalAddGroup E] [ContinuousConstSMul 𝕜 E] {p : Seminorm 𝕜 E} {r : ℝ},
0 < r → (Continuous ⇑p ↔ p.ball 0 r ∈ nhds 0)- Defined in
- Mathlib.Analysis.Seminorm
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 163 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites19
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- TopologicalSpacestatement and proof · cited by 24,529
- Modulestatement and proof · cited by 20,661
- AddCommGroupstatement and proof · cited by 12,871
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- Filterstatement · cited by 8,121
- nhdsstatement · cited by 5,554
- Continuousstatement and proof · cited by 2,592
- map_zeroproof · cited by 1,614
- IsTopologicalAddGroupstatement and proof · cited by 1,394
Cited by1
Results whose statement or proof uses this declaration.
- LinearMap.mem_span_iff_continuousproof · cited by 1