In this paper, we prove that (X, p) is separable if and only if there exists a w*-lower semicontinuous norm sequence \(\left\{ {{p_n}} \right\}_{n = 1}^\infty \) of (X*, p) such that (1) there exists a dense subset G_n of X* such that p_n is Gâteaux differentiable on G_n and dp_n (G_n) ⊂ X for all n ∊ N; (2) p_n ≤ p and p_n → p uniformly on each bounded subset of X*; (3) for any α ∈ (0, 1), there exists a ball-covering \(\left\{ {B\left( {x_{i,n}^*,{r_{i,n}}} \right)} \right\}_{i = 1}^\infty \) of (X*, p_n) such that it is α-off the origin and x ^*_{i, n} ∈ G_n. Moreover, we also prove that if X_i is a Gâteaux differentiability space, then there exist a real number α > 0 and a ball-covering \(\mathfrak{B}_i\) of X_i such that \(\mathfrak{B}_i\) is α-off the origin if and only if there exist a real number α > 0 and a ball-covering \(\mathfrak{B}\) of l^∞ (X_i) such that \(\mathfrak{B}\) is α-off the origin.

在本文中，我们证明（X，p）是可分离的，当且仅当存在w *-下半连续范数序列\（\ left \ {{{p_n}} \ right \} _ {n = 1} ^（X *，p）的\ infty \），使得（1）存在一个X *的密集子集G _n，使得p _n是在G _n和dp _n（G _n）X上对于所有n ∊ N可微的Gâteaux微分。; （2）p _Ñ ≤ p和p _Ñ → p统一在X *的每个有界子集上；（3）为任何α＆Element;（0,1），存在球覆盖\（\左\ {{B \左（{X_ {I，N} ^ *，{R_ {I，N}}} \右）} \右\} _ {i = 1} ^ \ infty \）的（X *，p _ñ），使得它是α -off原点和X ^* _I，N ∈ ģ _ñ。此外，我们还证明了如果X_我是糕点可微空间，则存在的实数α > 0和球覆盖\（\ mathfrak {B} _i \）的X_我使得\（\ mathfrak {B } _i \）是 α -off原点，当且仅当存在实数α > 0和球覆盖\（\ mathfrak {B} \）的升^∞（X_我），使得\（\ mathfrak {B} \）是α-偏离原点。