Existence of basis in some Whitney spaces

Existence of basis in locally convex spaces has been a hot subject in functional analysis for more than 40 years. We will give some partial solutions to this well-known problem. We will demonstrate two cases of Cantor-type sets with extension property under some special cases. These sets are KN with (DN) where lim sup αn < N and K∞ with (DN) where lim sup αn < ∞.