Theorems · Theorem · functional analysis
Seminorm.mem_ball
∀ {𝕜 : Type u_3} {E : Type u_7} [inst : SeminormedRing 𝕜] [inst_1 : AddCommGroup E] [inst_2 : SMul 𝕜 E]
(p : Seminorm 𝕜 E) {x y : E} {r : ℝ}, y ∈ p.ball x r ↔ p (y - x) < r- Defined in
- Mathlib.Analysis.Seminorm
- Cited by
- 10 results in Mathlib
- Foundations
- Depth 101 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- AddCommGroupstatement and proof · cited by 12,871
- SeminormedRingstatement and proof · cited by 446
- Seminormstatement and proof · cited by 272
- Seminorm.ballstatement · cited by 78
Cited by10
Results whose statement or proof uses this declaration.
- Seminorm.mem_ball_zeroproof · cited by 11
- Seminorm.ball_subset_closedBallproof · cited by 5
- Seminorm.convex_ballproof · cited by 4
- Seminorm.ball_eq_emptysetproof · cited by 2
- Seminorm.neg_ballproof · cited by 1
- Seminorm.ball_add_ball_subsetproof · cited by 1
- WithSeminorms.T1_of_separatingproof · cited by 1
- Seminorm.smul_ball_preimageproof · cited by 1
- Seminorm.absorbent_ballproof · cited by 1
- Seminorm.ball_smulproof · cited by 0