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Unformatted text preview: E Topological Vector Spaces Many of the important vector spaces in analysis have topologies that are generated by a family of seminorms instead of a given metric or a norm. We will consider these types of topologies in this appendix. References for the material in this appendix include the texts by Conway [Con90], Folland [Fol99], or Rudin [Rud91]. E.1 Motivation and Examples If X is a metric space then every open subset of X is, by definition, a union of open balls. The set of open balls is an example of a base for the topology on X (see Definition E.9). If X is also a vector space, it is usually very important to know whether these open balls are convex. This is certainly true if the metric is induced from a norm, but it is not true in general. Example E.1. Exercise B.57 tells us that if E R and 0 < p < 1 , then L p ( E ) is a complete metric space with respect to the metric d( f,g ) = bardbl f g bardbl p p . Be- cause the operations of vector addition and scalar multiplication are continu- ous on L p ( E ) , we call L p ( E ) a topological vector space (see Definition E.12). Unfortunately, the unit ball in L p ( E ) is not convex when p < 1 (compare the illustration in Figure E.1). In fact, it can be shown that L p ( E ) contains no nontrivial open convex subsets, so we cannot get around this issue by substi- tuting some other open sets for the open balls. When p < 1 , there is no base for the topology on L p ( E ) that consists of convex sets. For 0 < p < 1 , the topology on L p ( E ) is generated by a metric, but this metric is not induced from a norm or a family of seminorms. We will see that topologies that are generated from families of seminorms do have a base that consists of convex open sets. If the family of seminorms is finite, then we can find a single norm that induces the same topology. If the family of seminorms is countable, then we can find a single metric that induces the same topology. 386 E Topological Vector Spaces Minus 1.0 Minus 0.5 0.5 1.0 Minus 1.0 Minus 0.5 0.5 1.0 Minus 1.0 Minus 0.5 0.5 1.0 Minus 1.0 Minus 0.5 0.5 1.0 Minus 1.0 Minus 0.5 0.5 1.0 Minus 1.0 Minus 0.5 0.5 1.0 Minus 1.0 Minus 0.5 0.5 1.0 Minus 1.0 Minus 0.5 0.5 1.0 Fig. E.1. Unit circles in R 2 with respect to various metrics. Top left: 1 / 2 . Top right: 1 . Bottom left: 2 . Bottom right: . A significant advantage of having a topology induced from a metric is that the corresponding convergence criterion can be defined in terms of convergence of ordinary sequences instead of convergence of nets (see Section A.7). In this section we will give several examples of spaces whose topologies are induced from families of seminorms, indicating without proof some of the special features that we will consider in more detail in the following sections....
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