Abstract
The topology induced by a norm on a vector space is a very strong topology in the sense that it has many open sets. This has some advantages, especially since a function whose domain is such a space finds it particularly easy to be continuous, but it also has its disadvantages. For example, an infinite-dimensional normed space always has so many open sets that its closed unit ball cannot be compact. Because of this, many familiar facts about finite-dimensional normed spaces that are based on the Heine-Borel property cannot be immediately generalized to the infinite-dimensional case.
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© 1998 Springer Science+Business Media New York
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Megginson, R.E. (1998). The Weak and Weak Topologies. In: An Introduction to Banach Space Theory. Graduate Texts in Mathematics, vol 183. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0603-3_2
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DOI: https://doi.org/10.1007/978-1-4612-0603-3_2
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