\import Algebra.Algebra
\import Algebra.Module
\import Algebra.Module.LinearMap
\import Algebra.Monoid
\import Algebra.Ordered
\import Algebra.Ring
\import Arith.Real
\import Arith.Real.Field
\import Function.Meta
\import Logic
\import Logic.Meta
\import Meta
\import Order.LinearOrder
\import Order.PartialOrder
\import Order.StrictOrder
\import Paths
\import Paths.Meta
\import Topology.NormedAbGroup
\import Topology.NormedAlgebra
\import Topology.NormedModule
\import Topology.NormedRing
\open OrderedField

\class BanachSpace \extends CompleteNormedLModule {
\override R : NormedAAlgebra RealValuedRing
}

\lemma norm_*r {R : PseudoNormedCRing} {A : PseudoNormedAAlgebra R} {X : PseudoNormedLModule A} {a : R} {x : X} : norm (a *r x) = norm a * norm x
=> \case norm_zro_ide \with {
| byLeft p => norm_ide=0-module p *> inv Ring.zro_*-right *> pmap (_ *) (inv $norm_ide=0-module p) | byRight p => norm_*c *> pmap (* _) (pmap norm coefMap_*c *> norm_*c *> pmap (_ *) p *> ide-right) } \lemma norm_*r-ofPos {A : PseudoNormedAAlgebra RealValuedRing} {X : PseudoNormedLModule A} {a : Real} (a>=0 : 0 <= a) {x : X} : norm (a *r x) = a * norm x => norm_*r *> pmap (* _) (RealField.abs-ofPos a>=0) \record BoundedLinearMap \extends LinearMap, UniformNormedAbGroupMap { \override R : NormedAAlgebra RealValuedRing \override Dom : PseudoNormedLModule R \override Cod : PseudoNormedLModule R | isBounded : ∃ (C : Real) (0 < C) ∀ x (norm (func x) <= C * norm x) \default func-norm-uniform {eps} eps>0 => \case isBounded \with { | inP (C,C>0,h) => inP (pinv C>0 * eps, OrderedSemiring.<_*_positive_positive (pinv>0 C>0) eps>0, \lam {x} p => h x <∘r <_*_positive-right C>0 p <∘l Preorder.=_<= (inv *-assoc *> pmap (* _) (pinv-right C>0) *> ide-left)) } \default isBounded => \case func-norm-uniform RealAbGroup.zro<ide \with { | inP (delta,delta>0,h) => inP (pinv delta>0, pinv>0 delta>0, \lam x => <=_U-char.2 \lam {a} p => real_<_U.1 \have a>0 : 0 < Real.fromRat a => LinearlyOrderedSemiring.<=_*_positive_positive (LinearOrder.<_<=$ pinv>0 delta>0) norm>=0 <∘r real_<_U.2 p
\in transport2 (RealField.<) (inv (norm_*r-ofPos $LinearOrder.<_<= a>0) *> pmap norm (pmap (_ *r) func-*c *> inv *r-assoc *> pmap (*c _) (pmap coefMap (pinv-right a>0) *> AAlgebra.coefHom.func-ide) *> ide_*c)) ide-right$
RealField.<_*_positive-right a>0 $h {pinv a>0 *r x}$ transport2 (<) (*-comm *> inv (norm_*r-ofPos $LinearOrder.<_<=$ pinv>0 a>0)) (*-assoc *> pmap (_ *) (pinv-right a>0) *> ide-right) $<_*_positive-left (transport (< _) (inv *-assoc *> pmap (* _) (pinv-right delta>0) *> ide-left)$ RealField.<_*_positive-right delta>0 (real_<_U.2 p)) (pinv>0 a>0))
}
}