In mathematics, the Fraňková–Helly selection theorem is a generalisation of Helly's selection theorem for functions of bounded variation to the case of regulated functions. It was proved in 1991 by the Czech mathematician Dana Fraňková.
Background Let X be a separable Hilbert space, and let BV([0, T]; X) denote the normed vector space of all functions f : [0, T] → X with finite total variation over the interval [0, T], equipped with the total variation norm. It is well known that BV([0, T]; X) satisfies the compactness theorem known as Helly's selection theorem: given any sequence of functions (fn)n∈N in BV([0, T]; X) that is uniformly bounded in the total variation norm, there exists a subsequence
( f n ( k ) ) ⊆ ( f n ) ⊂ B V ( [ 0 , T ] ; X ) {\displaystyle \left(f_{n(k)}\right)\subseteq (f_{n})\subset \mathrm {BV} ([0,T];X)}
and a limit function f ∈ BV([0, T]; X) such that fn(k)(t) converges weakly in X to f(t) for every t ∈ [0, T]. That is, for every continuous linear functional λ ∈ X*,
λ ( f n ( k ) ( t ) ) → λ ( f ( t ) ) in X as k → ∞ . {\displaystyle \lambda \left(f_{n(k)}(t)\right)\to \lambda (f(t)){\mbox{ in }}X{\mbox{ as }}k\to \infty .}
Consider now the Banach space Reg([0, T]; X) of all regulated functions f : [0, T] → X, equipped with the supremum norm. Helly's theorem does not hold for the space Reg([0, T]; X): a counterexample is given by the sequence
f n ( t ) = sin ( n t ) . {\displaystyle f_{n}(t)=\sin(nt).}
One may ask, however, if a weaker selection theorem is true, and the Fraňková–Helly selection theorem is such a result.
Statement of the Fraňková–Helly selection theorem As before, let X be a separable Hilbert space and let Reg([0, T]; X) denote the space of regulated functions f : [0, T] → X, equipped with the supremum norm. Let (fn)n∈N be a sequence in Reg([0, T]; X) satisfying the following condition: for every ε > 0, there exists some Lε > 0 so that each fn may be approximated by a un ∈ BV([0, T]; X) satisfying
‖ f n − u n ‖ ∞ < ε {\displaystyle \|f_{n}-u_{n}\|_{\infty }<\varepsilon }
and
| u n ( 0 ) | + V a r ( u n ) ≤ L ε , {\displaystyle |u_{n}(0)|+\mathrm {Var} (u_{n})\leq L_{\varepsilon },}
where |-| denotes the norm in X and Var(u) denotes the variation of u, which is defined to be the supremum
sup Π ∑ j = 1 m | u ( t j ) − u ( t j − 1 ) | {\displaystyle \sup _{\Pi }\sum _{j=1}^{m}|u(t_{j})-u(t_{j-1})|}
over all partitions
Π = { 0 = t 0 < t 1 < ⋯ < t m = T , m ∈ N } {\displaystyle \Pi =\{0=t_{0}<t_{1}<\dots <t_{m}=T,m\in \mathbf {N} \}}
of [0, T]. Then there exists a subsequence
( f n ( k ) ) ⊆ ( f n ) ⊂ R e g ( [ 0 , T ] ; X ) {\displaystyle \left(f_{n(k)}\right)\subseteq (f_{n})\subset \mathrm {Reg} ([0,T];X)}
and a limit function f ∈ Reg([0, T]; X) such that fn(k)(t) converges weakly in X to f(t) for every t ∈ [0, T]. That is, for every continuous linear functional λ ∈ X*,
λ ( f n ( k ) ( t ) ) → λ ( f ( t ) ) in R as k → ∞ . {\displaystyle \lambda \left(f_{n(k)}(t)\right)\to \lambda (f(t)){\mbox{ in }}\mathbb {R} {\mbox{ as }}k\to \infty .}
References Fraňková, Dana (1991). "Regulated functions". Math. Bohem. 116 (1): 20–59. ISSN 0862-7959. MR 1100424.