On the Krein-Milman Property and the Bade Property

Throughout this paper a study on the Krein-Milmam Property and the Bade Property is entailed reaching the following conclusions: If a real topological vector space satisfies the Krein-Milmam Property, then it is Hausdorff; if a real topological vector space satisfies the Krein-Milmam Property and is locally convex and metrizable, then all of its closed infinite dimensional vector subspaces have uncountable dimension; if a real pseudo-normed space has the Bade Property, then it is Hausdorff as well but could allow closed infinite dimensional vector subspaces with countable dimension. On other hand, we show the existence of infinite dimensional closed subspaces of ℓ∞ with the Bade Property that are not the space of convergence associated to any series in a real topological vector space. Finally, we characterize unconditionally convergent series in real Banach spaces by means of a new concept called uniform convergence of series.