For a bounded subset \(A\) of a metric space \(X\), the Kuratowski measure of noncompactness is defined as \[ \alpha(A)= \inf\{\varepsilon>0: A\text{ can be covered by finitely many sets with diameter}<\varepsilon\} \] and the Hausdorff measure of noncompactness is defined as \[ \beta(A)= \inf\{\varepsilon>0: A\text{ can be covered by finitely many balls with diameter}<\varepsilon\}. \] An infinite bounded subset \(A\) of a metric space \(X\) is said to be \(\alpha\)-minimal (respectively \(\beta\)-minimal) if \(\alpha(A_0)= \alpha(A)\) (respectively \(\beta(A_0)= \beta(A)\)) for every infinite subset \(A_0\) of \(A\). The packing rates \(\delta\) and \(\delta'\) of \(X\) are defined as the supremum and the infimum (respectively) of the sets \[ \{\beta(A)/\alpha(A):A \text{ is a bounded, \(\alpha\) minimal and nonprecompact subset of }X\}. \] The quotient \(\delta/\delta'\) is denoted by \(\gamma\). This number \(\gamma\in [1,2]\) can be considered as a measure of packing in the space \(X\) and we can say that \(X\) has good packing if \(\gamma=1\) (this is the case of \(\ell_p\) spaces) and the worst packing occurs when \(\gamma=2\) (e.g. \(c_0\)).
In this paper the authors find the coefficients \(\delta(X)\), \(\delta'(X)\) and \(\gamma(X)\) when \(X\) is a direct sum of separable Banach spaces, with a monotonous norm (a norm in \(\mathbb{R}^k\) is said to be monotonous if \(|(a_1,a_2,\dots, a_k)|\leq|(b_1,b_2,\dots, b_k)|\) when \(0\leq a_i\leq b_i\) for every \(i=1,\dots,k\)) in the finite case and in the infinite case they only obtain results if the substitution space has a \(p\)-norm with \(1\leq p<\infty\). Main results:
(1) (Theorem 2.3): Let \(X_1,\dots,X_k\) be separable Banach spaces, \(|\cdot|\) a monotonous norm in \(\mathbb{R}^k\) and \(X_1\oplus\dots \oplus X_k\) the direct sum space with induced norm. Then \(\delta'(X_1\oplus\dots \oplus X_k)=\min_{1\leq i\leq k}\{\delta'(x_i)\}\), \(\delta(X_1\oplus\dots \oplus X_k)=\max_{1\leq i\leq k}\{\delta(X_i)\}\) and \(\gamma(X_1\oplus \dots\oplus X_k)= \frac{\max\{\delta(X_1),\dots, \delta(X_k)\}} {\min\{\delta' (X_1),\dots,\delta'(X_k)\}}\).
(2) (Theorem 2.11): Let \(\{X_k: k\in\mathbb{N}\}\) be a sequence of separable Banach spaces and \(1\leq p<\infty\). Then \(\delta'(\oplus_p X_k)= \inf_{k\in\mathbb{N}} \{2^{1-\frac{1}{p}}, \delta'(X_k)\}\), \(\delta(\oplus_p x_k)= \sup_{k\in\mathbb{N}} \{2^{1-\frac{1}{p}}, \delta(X_k)\}\), \(\gamma(\oplus_p X_k)=\frac {\sup_{k\in\mathbb{N}} \{2^{1-\frac{1}{p}}, \delta(X_k)\}} {\inf_{k\in\mathbb{N}} \{2^{1- \frac{1}{p}}, \delta'(X_k)\}}\).