Let \( \mathcal{P}(X,Y)\) denote the set of all continuous, linear projections from X onto its subspace Y. Put \(\begin{aligned} \lambda (Y,X)= inf\{ \Vert P\Vert : P \in \mathcal{P}(X,Y)\}. \end{aligned}\) A projection \(P_o \in \mathcal{P}(X,Y)\) is called a minimal projection if and only if \(\begin{aligned} \Vert P\Vert = \lambda (Y,X). \end{aligned}\) The aim of this paper is to give some estimates of \( \lambda (Y,X)\) in the case when \(\begin{aligned} X = \bigoplus _{j=1}^n X_j, \end{aligned}\) where \( (X_j, \Vert \cdot \Vert _j) \) for \(j=1,...,n,\) are Banach spaces and \(Y \subset X\) is a closed subspace of X equipped with a norm \(\begin{aligned} \Vert x\Vert = g(\Vert x_1\Vert _1,..., \Vert x_n\Vert _n), \end{aligned}\) where g is a monotone norm on \( \mathbb {K}^n, \) \(K= \mathbb {C}\) or \(K= \mathbb {C}.\) We present our results in a more general case of minimal extensions (see Definition (2)). This approach leads to some new estimates of norms of minimal extensions. Also the problems of unicity and strong unicity of minimal extensions (see Definition(3)) is studied.
