ABSTRACT
It is known (see (17.b) and (12.3)) that a LCS (X, P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3318.tif"/> ) is barrelled (resp. infrabarrelled, bornological) if and only if each lower semi–continuous seminorm on X (resp. lower semi–continuous seminorm on X which is bounded on bounded sets in X, seminorm on X which is bounded on bounded sets in X) is P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203749807/1e575708-9d79-4f43-a374-4d458ec8fff6/content/eq3319.tif"/> –Continuous. This section continues this idea of using seminorms satisfying some expected properties to investigate Schwartz spaces. This class of locally convex spaces can be regarded as the “best spaces in Analysis” since Schwartz spaces are closer to finite–dimensional spaces than Banach spaces are [bounded sets are precompact] on one hand, as well as are closer to Banach spaces than other classes of spaces are [complete Schwartz spaces are ultra–semi–reflexive].
