Theorems · Theorem · potential theory

InnerProductSpace.HarmonicOnNhd.mono

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] [inst_2 : FiniteDimensional ℝ E]
  {F : Type u_2} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace ℝ F] {f : E → F} {s t : Set E},
  InnerProductSpace.HarmonicOnNhd f s → t ⊆ s → InnerProductSpace.HarmonicOnNhd f t

If f is harmonic in a neighborhood of s, it is harmonic in a neighborhood of every subset.

Defined in: Mathlib.Analysis.InnerProductSpace.Harmonic.Basic
Cited by: 4 results in Mathlib
Foundations: Depth 245 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NormedAddCommGroup InnerProductSpace FiniteDimensional NormedAddCommGroup NormedSpace

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

Setstatement and proof · cited by 53,352
Realstatement and proof · cited by 25,697
NormedAddCommGroupstatement and proof · cited by 15,752
NormedSpacestatement and proof · cited by 12,499
InnerProductSpacestatement and proof · cited by 3,523
FiniteDimensionalstatement and proof · cited by 1,854
InnerProductSpace.HarmonicOnNhdstatement and proof · cited by 24

Cited by4

Results whose statement or proof uses this declaration.

InnerProductSpace.HarmonicContOnCl.circleAverage_re_herglotzRieszKernel_smulproof · cited by 2
InnerProductSpace.HarmonicContOnCl.circleAverage_eqproof · cited by 1
InnerProductSpace.HarmonicOnNhd.harmonicContOnClproof · cited by 1
InnerProductSpace.HarmonicContOnCl.monoproof · cited by 0