Abstract: This paper deals, constructively, with two theorems on the sequential continuity of linear mappings. The classical proofs of these theorems use the boundedness of the linear mappings, which is a constructively stronger property than sequential continuity; and constructively inadmissable versions of the Banach-Steinhaus theorem.

We show that the so-called weak Markov's principle (WMP) which states that every pseudo-positive real number is positive is underivable in T # :=EHA # +AC. Since T # allows to formalize (at least large parts of) Bishop's constructive mathematics this makes it unlikely that WMP can be proved within the framework of Bishop-style mathematics (which has been open for about 20 years). The underivability even holds if the ine#ective schema of full comprehension (in all types) for negated formulas (in particular for #-free formulas) is added which allows to derive the law of excluded middle for such formulas.

this paper we derive some results about sequentially continuous linear mappings within BISH. These results tend to reinforce our hope that such mappings may turn out to be bounded (continuous) after all. For background material on BISH, see [1], and for information about the relation between BISH, INT, and RUSS, see [2]. 2 Sequential continuity preserves Cauchyness

We consider notions of boundedness of subsets of the natural numbers N that occur when doing mathematics in the context of intuitionistic logic. We obtain a new characterization of the notion of a pseudobounded subset and formulate the closely related notion of a detachably finite subset. We establish metric equivalents for a subset of N to be detachably finite and to satisfy the ascending chain condition. Following Ishihara, we spell out the relationship between detachable finiteness and sequential continuity. Most of the results do not require countable choice.

A summability theorem of Landau, which classically is a simple consequence of the uniform boundedness theorem, is examined constructively.

In [Ish92] H. Ishihara introduced the so-called boundedness principle BD-N which claims that every countable pseudobounded subset of N is bounded. Here S ⊆ N is called pseudobounded iff for every sequence a ∈ SN there exists an n ∈ N such that ak < k for all k ≥ n. 1 Obviously, the principle BD-N is