Pointwise Weak Law of Large Numbers and Weak Law of Large Numbers in the norm topology of D[0,l] are shown to be equivalent under uniform convex tightness and uniform integrability conditions for weighted sums of a sequence of random elements in D[0,1]. Uniform convex tightness and uniform integrability conditions are jointly characterized. Marcinkiewicz–Zygmund–Kolmogorov's and Brunk– Chung's Strong Laws of Large Numbers are derived in the setting of D[0,l]-space under uniform convex tightness and uniform integrability conditions. Equivalence of pointwise convergence, convergence in the Skorokhod topology and convergence in the norm topology f o r sequences in D[0,l] is studied