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 ℝn 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.
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- 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 Bb(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 Bb is the space of bounded charges, but the pairing (Bb,ca) of Bb with finite measures is more common. See Section 11.6 for a discussion of this pair. Similarly, the dual of L∞ is larger than L1, but the pairing (L∞, L1) is more useful. This can be confusing at times. For instance, the τ(ℓ∞, ℓ1) Mackey topology for the dual pair (ℓ∞, ℓ1) 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.It is also customary to denote the value T( x ) by Tx.Google Scholar