We showed that the generalized contraction mapping defined on a closed convex subset of a weakly Cauchy normed space has a unique fixed point. Moreover, the sequence of iterates of any element in the domain of the given mapping is converging strongly to the fixed point of such a mapping

We give the sufficient conditions for the existence of a metric projection onto convex closed subsets of normed linear spaces which are reduced conditions than that in the case of reflexive Banach spaces and we find a general formula for the projections onto the maximal proper subspaces of the classical Banach spaces l p , 1 ≤ p < ∞ and c 0 . We also give the sufficient and necessary conditions for an infinite matrix to represent a projection operator from l p , 1 ≤ p < ∞ or c 0 onto anyone of their maximal proper subspaces.

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We find a lower estimation for the projection constant of the projective tensor product X⊗ ∧ Y and the injective tensor product X⊗ ∨ Y , we apply this estimation on some previous results, and we also introduce a new concept of the projection constants of operators rather than that defined for Banach spaces.