Abstract
We now set off on an expedition through the vast subject of functional analysis. No doubt the reader has some familiarity with this place, and will recognize some of the early landmarks of the journey. Our starting point is the study of normed spaces, which are situated at the confluence of two far-reaching mathematical abstractions: vector spaces, and topology. The setting is that of a vector space over the real numbers \({\mathbb{R}}\). The central idea of this chapter is that of a norm. It is studied in detail, along with the attendant concept of linear operators. The dual space is introduced. The chapter ends with a presentation of derivates, directional derivatives, tangent and normal vectors. These constructs, which allow one to reduce nonlinear situations to linear ones, will play a central role in later developments.
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Notes
- 1.
This inequality will be evident to us quite soon, once we learn that the function t↦ t p is convex on the interval (0,∞); see page 36.
- 2.
Rudin: “We relegate this distinction to the status of a tacit understanding.”
- 3.
Let Q be a subset of a partially ordered set (P, ⩽). A majorant of Q is an element p∈ P such that q ⩽ p ∀ q∈ Q. The set Q is totally ordered if every pair x,y of points in Q satisfies either x ⩽ y or y ⩽ x. P is inductive if every totally ordered subset Q of P admits a majorant. Zorn’s lemma affirms that every (nonempty) inductive partially ordered set P admits a maximal element: a point m such that x∈ P, m ⩽ x implies x = m.
- 4.
Strictly speaking, F′(x) is known as the Fréchet derivative, to distinguish it from (surprisingly many!) other objects of similar type.
- 5.
We have used the same notation DF(x) for the Jacobian as for the derivative of F. This is justified by the fact that the linear mapping from \({\mathbb{R}}^{ n}\) to \({\mathbb{R}}^{ k}\) which the matrix induces by matrix multiplication (on the left) is precisely the derivative of F at x.
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© 2013 Springer-Verlag London
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Clarke, F. (2013). Normed Spaces. In: Functional Analysis, Calculus of Variations and Optimal Control. Graduate Texts in Mathematics, vol 264. Springer, London. https://doi.org/10.1007/978-1-4471-4820-3_1
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DOI: https://doi.org/10.1007/978-1-4471-4820-3_1
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