1. A metrizable locally convex space X is stereotype if and only if X is complete. 2. Fr�chet spaces are locally convex spaces that are complete with respect to a translation invariant metric. 3. In particular, the following functional spaces frequently used in functional analysis are reflexive locally convex spaces : 4. A locally convex space carries a natural topology, called the initial topology induced by the seminorms. 5. If X is a pseudocomplete locally convex space , then its pseudosaturation X ^ \ vartriangle is stereotype. 6. |3 = As with any topological vector space, a locally convex space is also a uniform space. 7. Dually, if X is a pseudosaturated locally convex space , then its pseudocompletion X ^ \ triangledown is stereotype. 8. |4 = A Cauchy net in a locally convex space is a complete if and only if every Cauchy net converges. 9. The "'Bourbaki Alaoglu theorem "'is a generalization by Bourbaki to dual topologies on locally convex spaces . 10. However, the theory of locally convex spaces is not pre-supposed and the requisite material is developed in a short appendix.
