Let $E$ be a real linear space. A vectorial inner product is a mapping from $E\times E$ into a real ordered vector space $Y$ with the properties of a usual inner product. Here we consider $Y$ to be a $\mathcal B$-regular Yosida space, that is a Dedekind complete Yosida space such that $\bigcap _{J\in {\mathcal B}}J=\lbrace 0 \rbrace $, where $\mathcal B$ is the set of all hypermaximal bands in $Y$. In Theorem 2.1.1 we assert that any $\mathcal B$-regular Yosida space is Riesz isomorphic to the space $B(A)$ of all bounded real-valued mappings on a certain set $A$. Next we prove Bessel Inequality and Parseval Identity for a vectorial inner product with range in the $\mathcal B$-regular and norm complete Yosida algebra $(B(A),\sup _{\alpha \in A}|x(\alpha )|)$.
Let $k$ be a positive integer, and let $G$ be a simple graph with vertex set $V(G)$. A {\it $k$-dominating set} of the graph $G$ is a subset $D$ of $V(G)$ such that every vertex of $V(G)-D$ is adjacent to at least $k$ vertices in $D$. A {\it $k$-domatic partition} of $G$ is a partition of $V(G)$ into $k$-dominating sets. The maximum number of dominating sets in a $k$-domatic partition of $G$ is called the {\it $k$-domatic number} $d_k(G)$. \endgraf In this paper, we present upper and lower bounds for the $k$-domatic number, and we establish Nordhaus-Gaddum-type results. Some of our results extend those for the classical domatic number $d(G)=d_1(G)$.