Let $K_{m}-H$ be the graph obtained from $K_{m}$ by removing the edges set $E(H)$ of the graph $H$ ($H$ is a subgraph of $K_{m}$). We use the symbol $Z_4$ to denote $K_4-P_2.$ A sequence $S$ is potentially $K_{m}-H$-graphical if it has a realization containing a $K_{m}-H$ as a subgraph. Let $\sigma(K_{m}-H, n)$ denote the smallest degree sum such that every $n$-term graphical sequence $S$ with $\sigma(S)\geq \sigma(K_{m}-H, n)$ is potentially $K_{m}-H$-graphical. In this paper, we determine the values of $\sigma (K_{r+1}-Z, n)$ for $n\geq 5r+19, r+1 \geq k \geq 5,$ $j \geq 5$ where $Z$ is a graph on $k$ vertices and $j$ edges which contains a graph $Z_4$ but not contains a cycle on $4$ vertices. We also determine the values of $\sigma (K_{r+1}-Z_4, n)$, $\sigma (K_{r+1}-(K_4-e), n)$, $\sigma (K_{r+1}-K_4, n)$ for $n\geq 5r+16, r\geq 4$.

We propose a definition of generalized semi-inner products (g.s.i.p.). By relating them to duality mappings from a normed vector space to its dual space, a characterization for all g.s.i.p. satisfying this definition is obtained. We then study the Riesz representation of continuous linear functionals via g.s.i.p. As applications, we establish a representer theorem and characterization equation for the minimizer of a regularized learning from finite or infinite samples in Banach spaces of functions.

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Let$X$and$Y$be closed subspaces of the Lorentz sequence space$d(v, p)$and the Orlicz sequence space$l$_{$M$}, respectively. It is proved that every bounded linear operator from$X$to$Y$is compact whenever $$p > \beta _M : = \inf \{ q > 0:\inf \{ M(\lambda t)/M(\lambda )t^q :0< \lambda ,t \leqslant 1\} > 0.$$ As an application, the reflexivity of the space of bounded linear operators acting from$d(v, p)$to$l$_{$M$}is characterized.