Several theorems on boundedness and equicontinuity

This paper presents several results concerning equicontinuity of a pointwise-bounded family of linear transformations on a Banach space. The first is the following generalization of the Banach-Steinhaus Theorem: Let $\{ {T_\alpha }|\alpha \in A\}$ be a pointwise-bounded family of linear transformations from a Banach space $X$ to a normed linear space $Y$, and assume that, for each $\alpha \in A,{T_\alpha }$ is continuous on a closed subspace ${S_\alpha }$ of $X$. Then $\exists {\alpha _1}, \cdots ,{\alpha _n} \in A$ such that the family is equicontinuous on $\bigcap \nolimits _{k = 1}^n {{S_{\alpha k}}}$. The second theorem deals with a pointwise-bounded family of linear transformations from a Banach space $X$ to a normed linear space with a continuous bilinear mapping into another normed linear space. The others deal with homomorphisms of Banach algebras.