English
The gauge rescale map gaugeRescale s t sends each x to the point on the same ray as x whose gauge with respect to t matches the gauge of x with respect to s. In formula, gaugeRescale s t x = (gauge s x / gauge t x) · x.
Русский
Пере масштаб gauge-проекции gaugeRescale s t отправляет каждую точку x на ту же лучевую величину, чтобы gauge_t(x) совпал с gauge_s(x). Теперe gaugeRescale s t x = (gauge s x / gauge t x) · x.
LaTeX
$$$\\operatorname{gaugeRescale}(s,t,x) = \\left(\\dfrac{\\operatorname{gauge}(s,x)}{\\operatorname{gauge}(t,x)}\\right) \\cdot x$$$
Lean4
/-- The gauge rescale map `gaugeRescale s t` sends each point `x` to the point `y` on the same ray
that has the same gauge w.r.t. `t` as `x` has w.r.t. `s`.
The characteristic property is satisfied if `gauge t x ≠ 0`, see `gauge_gaugeRescale'`.
In particular, it is satisfied for all `x`,
provided that `t` is absorbent and von Neumann bounded. -/
def gaugeRescale (s t : Set E) (x : E) : E :=
(gauge s x / gauge t x) • x
