Infinite-dimensional calculus with a view towards Lie theory Helge - PDF document

Infinite-dimensional calculus with a view towards Lie theory Helge Gl¨ ockner (Universit¨ at Paderborn) Hamburg, February 16, 2015

Overview § 1 Basics of infinite-dimensional calculus § 2 Inverse functions and implicit functions § 3 Exponential laws for function spaces § 4 Non-linear maps on locally convex direct limits § 5 Measurable regularity and applications

§ 1 Basics of ∞ -dim calculus Defn. E , F locally convex spaces, U ⊆ E open. A map f : U → F is called C 1 if it is continuous, the directional derivatives � f ( x, y ) := ( D y f )( x ) = d � d f ( x + ty ) � dt � t =0 exist for all x ∈ U , y ∈ E , and the map f : U × E → F d is continuous. The map f is called C k with k ∈ N 0 ∪ {∞} if the iterated directional derivatives d j f ( x, y 1 , . . . , y j ) := ( D y j · · · D y 1 f )( x ) exist for all j ∈ N 0 such that j ≤ k and define continuous functions d j f : U × E j → F. Rem f is C k +1 iff f is C 1 and d f : U × E → F is C k . C ∞ -maps are also called smooth .

Basic facts (a) d f ( x, . ): E → F is linear (b) The Chain Rule holds: If f : U → V and g : V → F are C k , then also g ◦ f : U → F is C k , with d ( g ◦ f )( x, y ) = dg ( f ( x ) , d f ( x, y )) . Defn. Smooth manifolds modelled on locally convex TVS E are defined as usual: Hausdorff topological space M with an atlas of homeomorphisms φ : M ⊇ U → V ⊆ E (”charts”) between open sets such that the chart changes are smooth. Defn. Lie group = group G , equipped with a smooth manifold structure modelled on a locally convex space such that the group operations are smooth maps. L ( G ) := T e G , with Lie bracket arising from the identification of y ∈ L ( G ) with the correspond- ing left invariant vector field.

Comparison with other approaches to differential calculus The approach to ∞ -dimensional calculus pre- sented here goes back to A. Bastiani and is also known under the name of Keller’s C k c -theory . Classical calculus in Banach spaces A map f : E ⊇ U → F between Banach spaces echet differentiable ( FC 1 ) is called continuously Fr´ if it is totally differentiable and f ′ : U → ( L ( E, F ) , � . � op ) is continuous. If f is FC 1 and f ′ is FC k , then f is called FC k +1 . Fact: f is C k +1 ⇒ f is FC k ⇒ f is C k Convenient differential calculus If E is a Fr´ echet space, then a map f : E ⊇ U → F is C ∞ iff f ◦ γ : R → F is C ∞ for each C ∞ -curve γ : R → U , i.e., iff f is smooth in the sense of the convenient differential calculus (developed by Fr¨ olicher, Kriegl and Michor).

Likewise if E is a Silva space (or (DFS)-space), i.e., a locally convex direct limit E = lim → E n of Banach spaces E 1 ⊆ E 2 ⊆ · · · such that all inclusion maps E n → E n +1 are compact operators. Beyond metrizable or Silva domains, the smooth maps of convenient differential calculus need not be C ∞ in the sense used here (they need not even be continuous). Diffeological spaces If E is a Fr´ echet space or a Silva space, then a map f : E ⊇ U → F is C ∞ if and only if f ◦ γ : R n → F is C ∞ for each n ∈ N and C ∞ - map γ : R n → U (and it suffices to take n = 1 as already mentioned).

Main classes of ∞ -dim Lie groups Linear Lie groups G ≤ A × Mapping groups Diffeomorphism groups e.g. C ∞ ( M, H ) Diff( M ) M compact Direct limit groups G = � n G n with G 1 ≤ G 2 ≤ · · · fin-dim Here A is a Banach algebra or a continuous inverse algebra (CIA) A × is open and A × → A , x �→ x − 1 is continuous

Elementary facts for f : E ⊇ U → F . (a) If f ( U ) ⊆ F 0 for a closed vector subspace F 0 ⊆ F , then f is C k iff f | F 0 is C k j ∈ J F j , then f is C k iff each of its (b) If F = � components f j is C k . (c) If F = lim ← F n for a projective sequence · · · → F 2 → F 1 , then f is C k iff π n ◦ f is C k for each n ∈ N , where π n : F → F n is the limit map. E.g. C ∞ ([0 , 1] , R ) = lim ← C n ([0 , 1] , R ) for n ∈ N ; C k +1 ([0 , 1] , R ) → C ([0 , 1] , R ) × C k ([0 , 1] , E ) , γ �→ ( γ, γ ′ ) linear topological embedding, closed image. Hence a map f to C ∞ ([0 , 1] , R ) is smooth iff it is smooth as a map to C k ([0 , 1] , R ) for each finite k . A map to C k +1 ([0 , 1] , R ) is smooth iff it is smooth as a map to C ([0 , 1] , R ) and x �→ f ( x ) ′ is smooth as a map to C k ([0 , 1] , R ) ❀ simple inductive proofs for smoothness of maps to function spaces

Mean Value Theorem. If f : E ⊇ U → F is C 1 and x, y ∈ U such that x + [0 , 1]( y − x ) ⊆ U , then � 1 f ( y ) − f ( x ) = 0 d f ( x + t ( y − x ) , y − x ) dt. Defn. Let E be a locally convex space. A (nec. unique) element z ∈ E is called the weak integral of a continuous path γ : [ a, b ] → E if � b for all λ ∈ E ′ . λ ( z ) = a λ ( γ ( t )) dt � b Write a γ ( t ) dt := z . Mappings on non-open sets: Let U ⊆ E be a subset with dense interior which is locally convex , i.e., each x ∈ U has a relatively open, convex neighbourhood in U . Say that a con- tinuous map f : U → F is C k if f | U 0 is C k and d j ( f | U 0 ): U 0 × E j → F extends to a continuous map d j f : U × E j → F for each j ∈ N such that j ≤ k .

If f : E ⊇ U → F , then the directional difference quotients f ( x + ty ) − f ( x ) t make sense for all ( x, y, t ) in the set U [1] := { ( x, y, t ) ∈ U × E × R : x + ty ∈ U } such that t � = 0. Fact. A continuous map f is C 1 if and only if there is a continuous map f [1] : U [1] → F with f [1] ( x, y, t ) = f ( x + ty ) − f ( x ) t or all ( x, y, t ) ∈ U [1] such that t � = 0. f ( x, y ) = lim t → 0 f [1] ( x, y, t ) = f [1] ( x, y, 0) Indeed, d in this case and thus f is C 1 . If f is C 1 , define � f ( x + ty ) − f ( x ) if t � = 0; f [1] ( x, y, t ) := t f ( x, y ) if t = 0. d By the Mean Value Theorem, for | t | small have � 1 f [1] ( x, y, t ) = f ( x + sty, y ) ds. 0 d Since weak integrals depend continuously on parameters, f [1] is continuous.

First application of f [1] : Very easy proof of the Chain Rule. Another application, with a view towards the commutator formula: If G is a Lie group and γ 1 , γ 2 ∈ C 1 ([0 , r ] , G ) with γ 1 (0) = γ 2 (0) = e , then η : [0 , r 2 ] → G , √ √ √ √ t ) − 1 γ 2 ( t ) − 1 η ( t ) := γ 1 ( t ) γ 2 ( t ) γ 1 ( is C 1 . Proof. η is C 1 on ]0 , r 2 ]. We show ( η | ]0 ,r 2 ] ) ′ has a continuous extension to [0 , r 2 ]. Let U ⊆ G , V ⊆ U be open identity neighbour- hoods with V V V − 1 V − 1 ⊆ U . Identify U with an open set in E using a chart, such that e = 0. The map f ( x, y ) := xyx − 1 y − 1 f : V × V → U, is smooth with d f (0 , 0 , v, w ) = 0 and d 2 f (0 , 0; x, y ; x, y ) = 2[ x, y ] .

The assertion now follows with a lemma by K.-H. Neeb: Lemma If U ⊆ E is open, γ : [0 , 1] → U is C 1 and f : U → F a C 2 -map with d f ( γ (0) , . ) = 0 , then √ η : [0 , 1] → U, t �→ f ( γ ( t )) is C 1 with η ′ (0) = 1 2 d 2 f ( γ (0) , γ ′ (0) , γ ′ (0)) . Proof: We may assume that γ (0) = 0 and f (0) = 0. Noting that =0 √ √ � �� � √ √ √ √ t γ ( t ) − γ ( 0) t γ [1] (0 , 1 , √ γ ( t ) = = t ) , t we get for t > 0 √ √ √ 1 t )) − 1 η ′ ( t ) t ); γ ′ ( f (0 , γ ′ ( = √ f ( γ ( √ t )) td td 2 2 � �� � =0 √ √ √ 1 f ) [1] (0 , γ ′ ( t ); γ [1] (0 , 1 , = 2( d t ) , 0; t ) The right-hand-side makes sense also for t = 0 and is continuous on [0 , 1]. Hence η is C 1 , with 1 η ′ (0) f ) [1] (0 , γ ′ (0); γ ′ (0) , 0; 0) = 2( d 1 2 d 2 f (0 , γ ′ (0) , γ ′ (0)) . =

Literature for § 1: • A. Bastiani, Applications diff´ erentiables et erentiables de dimension infinie , vari´ et´ es diff´ 1964. • W. Bertram, HG, and K.-H. Neeb, Differ- ential calculus over general base fields and rings , 2004. • Cartan, H., “Calcul diff´ erentiel,” 1967. • HG, Infinite-dimensional Lie groups with- out completeness restrictions , 2002. • HG and K.-H. Neeb, ”Infinite-Dimensional Lie Groups,” book in preparation. • H. H. Keller, “Differential Calculus in Lo- cally Convex Spaces,” 1974. • A. Kriegl and P. W. Michor, “The Conve- nient Setting of Global Analysis,” 1997. • J. Milnor, Remarks on infinite-dimensional Lie groups , 1984. • K.-H. Neeb, Towards a Lie theory of locally convex groups , 2006.

§ 2 Inverse functions and implicit functions Implicit Function Theorem (HG’05) Let E be a locally convex space, F be a Banach space, G ⊆ E × F be open, ( p 0 , y 0 ) ∈ G and f : G → F be a C k -map such that f ( p 0 , y 0 ) = 0 and f p 0 : y �→ f ( p 0 , y ) has invertible differential at y 0 . If F has finite dimension, assume k ≥ 1 ; otherwise, assume that k ≥ 2 . Then there exist open neighbour- hood P ⊆ E of p 0 and V ⊆ F of y 0 such that { ( p, y ) ∈ P × V : f ( p, y ) = 0 } = graph( φ ) for a C k -function φ : P → V . (Compare Hiltunen 1999, Teichmann 2001 for related results in other settings of ∞ -dim calculus)