About absolutely summing operators

Question

The concept of absolute summing operator between two Banach spaces is already well-known. I just would like to know if one can extend the definition between general spaces, say locally convex spaces. I would like to be clarified with this and thanks for your help in advance...

Answer 1

The notion of absolutely summing mappings for locally convex spaces was given by Albrecht Pietsch in the book Nuclear locally convex spaces. p. 34.

The definition is exactly the same as for normed spaces: operator $T$ is called absolutely summing if for any convergent series $\sum_{i} x_i$, the series $\sum_iT(x_i)$ is absolutely convergent.

The only thing in this definition you need to redefine for locally convex spaces is absolute convergence.

We say that the series $\sum_{i}x_i$ in a locally convex space is absolutely convergent if for any neighbourhood of zero the series $\sum_{i} p_U(x_i)$ converges in $\mathbb{R}$ (here $p_U$ denote Minkowski functional of $U$).