Infinite Dimensional Analysis pp 181-205 | Cite as

# Normed spaces

Chapter

## Abstract

This chapter studies some of the special properties of normed spaces. All finite dimensional spaces have a natural norm, the Euclidean norm. On a finite dimensional vector space, the Hausdorif linear topology the norm generates is unique (Theorem 4.61). The Euclidean norm makes ℝ^{n} into a complete metric space. A normed space that is complete in the metric induced by its norm is called a *Banach space*. Here is an overview of some of the more salient results in this chapter.

## Keywords

Banach Space Normed Space Weak Topology Dual Pair Norm Topology
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## References

- 1.The natural duality of a normed space with its norm dual is not always the most useful pairing. Two important examples are the normed spaces B
_{b}(Х) of bounded Borel measurable functions on a metrizable space, and the space L_{∞}(μ) of μ-integrable functions. (Both include ℓ_{∞}as a special case.) The dual of B_{b}is the space of bounded charges, but the pairing (B_{b},ca) of B_{b}with finite measures is more common. See Section 11.6 for a discussion of this pair. Similarly, the dual of L_{∞}is larger than L_{1}, but the pairing (L_{∞}, L_{1}) is more useful. This can be confusing at times. For instance, the τ(ℓ_{∞}, ℓ_{1}) Mackey topology for the dual pair (ℓ_{∞}, ℓ_{1}) is not the norm topology on ℓ_{∞}: it is weaker. In this chapter at least, we do not deal with other pairings. But when it comes to applying these theorems, make sure you know your dual.Google Scholar - 2.

## Copyright information

© Springer-Verlag Berlin Heidelberg 1994