ABSTRACT
A locally convex linear space X $ \mathcal{X} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315151601/40b40f18-f94e-4633-83f0-b519efc5b015/content/inline-math9_1.tif"/> is a generalization of a normed space, the topology on X $ \mathcal{X} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315151601/40b40f18-f94e-4633-83f0-b519efc5b015/content/inline-math9_2.tif"/> given by a family of seminorms rather than a single norm. These spaces occur in a variety of contexts, including operator theory and distributions. In the present chapter we develop the properties of locally convex spaces to a sufficient extent that will allow the discussion of weak and weak ∗ $ ^* $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315151601/40b40f18-f94e-4633-83f0-b519efc5b015/content/inline-math9_3.tif"/> topologies in the next chapter and the material on distributions in Chapter 15 to be seen from a general vantage point. Additional properties of locally convex spaces as well as applications are considered in Chapter 14.
