AN EXTENSION OF DINI’S LEMMA TO NETS OF FUNCTIONS INTO UNIFORM SPACES

Bhamini M. P. Nayar

A classical and important lemma of Dini in analysis states:

\underline{Theorem D}. If \{f_{n}\} is a sequence of real–valued continuous functions on [0, 1] such that {f_{n}(x)} is a non–increasing sequence converging to 0 for each x ∈ [0, 1] then f_{n} → 0 uniformly.

In this note we extend this lemma to nets of functions mapping into uniform spaces. The following readily established results are used as patterns to formulate and prove the extension.

(1^{∘}) A non–increasing sequence {x_{n}} of nonnegative reals converges to 0 if and only if for each ϵ > 0, ∅ ≠ {n : x_{n} − 0 < ϵ} ⊂ {n : x_{k} − 0 < ϵ for every k ≥ n}.

An equivalent form of Theorem D is

(2^{∘}) If {f_{n}} is a sequence of real–valued continuous functions on [0, 1] such that {|f_{n}(x)|} is a non–increasing sequence converging to 0 for each x ∈ [0, 1] then f_{n} → 0 uniformly.

These observations motivate the following definitions.

\underline{Definition 1}. If X is a uniform space with uniformity U we say a net \{x_{n}\}_{Λ} converges to x ∈ X in a monotone fashion if for each U ∈ U, ∅ ≠ {n : (x_{n}, x) ∈ U} ⊂ {n : (x_{k}, x) ∈ U for every k ∈ Λ with k ≥ n}.

\underline{Definition 2}. Let Y be a uniform space with uniformity U, X be a nonempty set, and f: X → Y a function. We say that a net {f_{n}}_{Λ} of functions from X to Y is pointwise monotonically convergent to f if {f_{n}(x)}_{Λ} converges to f(x) in a monotone fashion for each x ∈ X.

A net {f_{n}(x)}_{Λ} from a topological space X to a uniform space Y with uniformity U converges to f: X → Y uniformly if for each U ∈ U, (f_{n}(x), f(x)) ∈ U ultimately. For facts on topological and uniform spaces used here without definition see [1].

We are now prepared to present our extension of Dini’s lemma.

\underline{Theorem}. Let X be a compact space and let Y be a uniform space with uniformity U. Let {f_{n}}_{Λ} be a net of continuous functions from X to Y. If {f_{n}}_{Λ} is pointwise monotonically convergent to the continuous function f: X → Y then {f_{n}}_{Λ} converges to f uniformly.

\underline{Proof}. Suppose \{f_{n}\}_{Λ} does not converge uniformly to f. Choose a cofinal △ ⊂ Λ, a net {x_{n}}_{△} in X and a U ∈ U such that each n ∈ △ satisfies (f_{n}(x_{n}), f(x_{n})) ∉ U. Choose a symmetric V ∈ U such that V ∘ V ∘ V = V^{3} ⊂ U. Since X is compact choose x ∈ X and a subnet of {x_{n}}_{△}, called again {x_{n}}_{△}, such that x_{n} → x. Since f_{n}(x) → f(x), choose m ∈ △ such that (f_{m}(x), f(x)) ∈ V. Since f, f_{m} are continuous at x choose an open set A in X such that x ∈ A, (f_{m}(p), f_{m}(x)) ∈ V, (f(p), f(x)) ∈ V, for all p ∈ A. Choose k ∈ △ such that k > m and x_{k} ∈ A. Then (f_{m}(x), f(x)) ∈ V, (f_{m}(x_{k}), f_{m}(x)) ∈ V, (f(x_{k}), f(x)) ∈ V. Therefore (f_{m}(x_{k}), f(x_{k})) ∈ V^{3} ⊂ U. Since {f_{n}(x_{k})}_{Λ} converges to f(x_{k}) in a monotone fashion and k > m it follows that f_{k}(x_{k}), f(x_{k})) ∈ V^{3} ⊂ U, a contradiction. The proof is complete.

\underline{Reference}

1. I. M. James, Topological and Uniform Spaces, Springer–Verlag, New York, 1987.

Bhamini M. P. Nayar Mathematics Department Morgan State University Baltimore, MD 21251 email: bnayar@morgan.edu