The semi Orlicz space \(cs\cap d_{1}\) (Q2853295)

An Orlicz function is a mapping \(M:[0,\infty)\to[0,\infty)\) which is continuous, non-decreasing and convex with \(M(0)=0\), \(M(x)>0\) for \(x>0\) and \(M(x)\to\infty\) for \(x\to\infty\). If convexity is replaced by sub-additivity (\(M(x+y)\leq M(x)+M(y)\)), then the function is called a modulus function.NEWLINENEWLINE Let \(\omega\) be the set of all complex sequences, consider the difference operator \(\Delta:\omega\to \omega\) defined for \(x\in \omega\) by \(\Delta x=(x_k-x_{k+1})_{k=0}^\infty\), and introduce NEWLINE\[NEWLINE \begin{aligned} \Gamma_M&=\Bigl\{x\in \omega:\lim_{k\to\infty}M\left(| x_k| ^{1/k}/\rho\right)=0\text{ for some }\rho>0\Bigr\}, \\ \Lambda_M&=\Bigl\{x\in \omega:\sup_k M\left(| x_k| ^{1/k}/\rho\right)=0\text{ for some }\rho>0\Bigr\}. \end{aligned} NEWLINE\]NEWLINE Furthermore, let \(\Gamma_M(\Delta)=\{x\in \omega:\Delta x\in \Gamma_M\}\) and \(\Lambda_M(\Delta)=\{x\in \omega:\Delta x\in\Lambda_M\}\); these spaces are metric spaces with respect to NEWLINE\[NEWLINE d(x,y)=\inf\Bigl\{\rho>0:\sup_k M\left(| \Delta x_k-\Delta y_k| ^{1/k}/\rho\right)\leq1\Bigr\}. NEWLINE\]NEWLINE The main results are then given by 11 propositions, of which the first three are given below:NEWLINENEWLINE Proposition 3.1: \(\Gamma\subset\Gamma_M(\Delta)\).NEWLINENEWLINE Proposition 3.2: \(\Gamma_M\subset\Gamma_M(\Delta)\) and the inclusion is strict.NEWLINENEWLINE Proposition 3.3: Define \(d_1=\{a=(a_k)\in \omega:\sup_{n,k\in\mathbf N}\big| \sum_{j=0}^k\sum_{i=j}^na_i\big| <\infty\}\); then \(| \Gamma_M(\Delta)| ^\beta=cs\cap d_1\). \big(For a sequence space \(X\), we have \(X^\beta=\big\{a=(a_k):\sum_{k=1}^\infty a_kx_k \text{ converges for each }x\in X\big\}\).\big)NEWLINENEWLINE After this, four other propositions are stated in Section 3, followed in Section 4 by four propositions on properties of semi difference Orlicz spaces \(cs\cap d_1\).NEWLINENEWLINE \smallskip In the introduction, the authors state that ``\dots we introduce a new class of sequence spaces namely the semi difference Orlicz space \(cs\cap d_1\)''; now Proposition 3.3 is the first occurrence of the set \(cs\cap d_1\), and in the proof the symbol \(cs\) appears, but there is not given a formal definition of this \(cs\), so it is not yet clear what is meant by the authors.NEWLINENEWLINE Definition 4.1 is the closest thing: an FK-space (locally convex Fréchet space with the property that the coordinate projections are continuous) is called ``semi difference Orlicz space \(cs\cap d_1\)'' if its dual \((\Delta X)^f\subset cs\cap d_1\), but this only comes near the end of the paper.