DLM in normed carriers #
Over finite-dimensional real normed carriers the production map is continuous, hence Lipschitz
with constant its operator norm: meanings within ε of each other produce forms within
‖production‖ * ε of each other. This is the quantitative form of the papers' form–meaning
isomorphy claims ([chuang-bell-tseng-baayen-2026], [lu-chuang-baayen-2026]).
References #
- [Y.-Y. Chuang, M. J. Bell, Y.-H. Tseng and R. H. Baayen, Word-specific tonal realizations in Mandarin (2026)][chuang-bell-tseng-baayen-2026]
- [Y. Lu, Y.-Y. Chuang and R. H. Baayen, The realization of tones in spontaneous spoken Taiwan Mandarin (2026)][lu-chuang-baayen-2026]
theorem
DiscriminativeLexicon.Linear.lipschitzWith_production
{F : Type u_1}
{M : Type u_2}
[NormedAddCommGroup F]
[NormedAddCommGroup M]
[NormedSpace ℝ F]
[NormedSpace ℝ M]
[FiniteDimensional ℝ M]
(D : Linear ℝ F M)
:
LipschitzWith ‖LinearMap.toContinuousLinearMap D.production‖₊ ⇑D.production
The production map is Lipschitz with constant its operator norm.
theorem
DiscriminativeLexicon.Linear.norm_production_sub_le
{F : Type u_1}
{M : Type u_2}
[NormedAddCommGroup F]
[NormedAddCommGroup M]
[NormedSpace ℝ F]
[NormedSpace ℝ M]
[FiniteDimensional ℝ M]
(D : Linear ℝ F M)
{e₁ e₂ : M}
{ε : ℝ}
(h : ‖e₁ - e₂‖ ≤ ε)
:
Meanings within ε of each other produce forms within ‖production‖ * ε.