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Let H be the set of all sequences h=(h k ) k=1 ∞ of Os and 1s. A sequence x in a topological sequence space E has the property of absolute boundedness |AB| if H.x={y|y k =h k x k , h∈H} is a bounded subset of E. The subspace E |AB| of all sequences with absolute boundedness in E has a natural topology stronger than that induced by E. A sequence x has the property of absolute sectional convergence |AK| if, under this stronger topology, the net {h.x} converges to x, where h ranges over all sequences in H with a finite number of 1s ordered coordinatewise (h'≤h″ iff ∀k, h' k ≤h″ k ). Absolute boundedness and absolute convergence are investigated. It is shown that, for an FK-space E, we have E=E |AB| if and only if E=l ∞.E, and every element of E has the property |AK| if and only if E=c a .E

Absolute boundedness and absolute convergence in sequence spaces