Sequentially Continuity in Constructive Mathematics

Abstract

The classical validity of many important theorems of functional analysis, such as the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, depends on Baire’s theorem about complete metric spaces, which is an indispensable tool in this area. A form of Baire’s theorem has a constructive proof [5, Theorem 1.3], but its classical equivalent,

if a complete metric space is the union of a sequence of its subsets, then the closure of at least one set in the sequence must have nonempty interior

which is used in the standard argument to prove that the above theorems have no known constructive proof. If we could prove the Baire’s theorem of the above form, we would have the following forms of constructive versions of Banach’s inverse mapping theorem, the open mapping theorem, the closed graph theorem, the Banach-Steinhaus theorem and the Hellinger-Toeplits theorem:

• Theorem 1 (Banach’s inverse mapping theorem) LetT be a one-one continuous linear mapping of a separable Banach space E onto a Banach space F. Then T^-1 is continuous.

• Theorem 2 (The open mapping theorem) Let T be a continuous linear mapping of a Banach space E onto a Banach space F such that ker(T) is located¹. Then T is open.

• Theorem 3 (The closed graph theorem) Let T be a linear mapping of a Banach space E into a Banach space F such that graph(T) is closed and separable. Then T is continuous.

• Theorem 4 (The Banach-Steinhaus theorem) Let {Tm} be a sequence of continuous linear mappings from a separable Banach space E into a normed space F such that

  $$ Tx: = \mathop {\lim }\limits_{m \to \infty } {T_m}x $$

  exists for all x ∈ E. Then T is continuous.

• Theorem 5 (The Hellinger-Toeplitz theorem) Let T be a linear mapping from a Banach space E into a separable normed space with the following property: if f is a normable² linear functional f on F, and {x_n} converges to 0 in E, then f (Tx_n) → 0. Then T is continuous.