locally convex space sentence in Hindi

Examples

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.