If (X,∥.∥) is a real normed lattice, then p(x)=∥x+∥ defines an asymmetric norm on X. We study the convergence of sequences in the asymmetrically normed lattice (X,p) and give a characterization of the set of limit points of a convergent sequence in the case X=ℝm. These results enable us to prove the left-K-sequential completeness of the asymmetrically normed lattices ℝm, C(Ω), c0, ℓ∞ and ℓp (1≦p<∞).
L. M. García-Raffi, S. Romaguera and E. A. Sánchez Pérez, Sequence spaces and asymmetric norms in the theory of computational complexity, Math. Comput. Model., 36 (2002), 717–728.