Definition:Equicontinuous Family of Linear Transformations between Topological Vector Spaces

Definition

Let $K$ be a topological field.

Let $X$ be a topological vector space over $K$.

Let $\Gamma = \family {T_\alpha}_{\alpha \mathop \in I}$ be a set of linear transformations $T_\alpha : X \to Y$.

We say that $\Gamma$ is equicontinuous if and only if:

for each open neighborhood $W$ of $\mathbf 0_Y$, there exists an open neighborhood $V$ of $\mathbf 0_X$ such that:

$T_\alpha \sqbrk V \subseteq W$ for each $\alpha \in I$.

Also see

• Results about equicontinuous families of linear transformations between topological vector spaces can be found here.

Sources

• 1991: Walter Rudin: Functional Analysis (2nd ed.) ... (previous) ... (next): $2.3$: Equicontinuity