Manuscript En08-B@1 · En08-B.tex · corollary · 23cac52a

Corollary: Uniform endpoint operational limit

The connecting theorem › The precise comparison topology · lines 952–974 · proof included or linked

En08-B@1#cor:operational-limit, En08-B@1#eq:operational-limit

Under the assumptions of Theorem thm:main(iv),

∥ x ∥_{2} = c T \leq L sup Err_{j} (T, x) \leq \frac{ε _{j} + c B _{j} ( L )}{ℓ} + min {1, α_{j} + H_{j} (L) e_{j} + β_{j}} .

The right-hand side tends to zero whenever $α_{j}, β_{j} \to 0$ and $j e_{j} \to 0$ . In the ideal construction, the channel-error term is zero at every finite refinement.

Proof. Substitute the three estimates from Theorem thm:main into (eq:operational-error). Equations (eq:B-expanded) and (eq:H-expanded) give the stated convergence.

Displayed formulas in this passage

En08-B@1#eq:operational-limit · line 955
$∥ x ∥_{2} = c T \leq L sup Err_{j} (T, x) \leq \frac{ε _{j} + c B _{j} ( L )}{ℓ} + min {1, α_{j} + H_{j} (L) e_{j} + β_{j}} .$

Corollary: Uniform endpoint operational limit

Displayed formulas in this passage

Nearby in this section