Another point that makes the text introductory is the use of an essentially uniform mathematical language and way of thinking, one which is no doubt familiar from elementary lectures in analysis that did not worry much about its connections with algebra and topology. Of course we shall use some elementary topological concepts, which may be new, but in fact only a few remarks here and there pertain to algebraic or differential topological concepts and methods.
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Buy Softcover. FAQ Policy. It covers basic real analysis, Lebesgue measure and integral in N-dimensional real spaces.
It also covers basic m-differentiable function spaces C. There are no proofs or exercises in this chapter.
Chapter 2 — Normed Vector Spaces — the start of linear functional analysis. This chapter covers continuous linear or multilinear operators, compact linear operators, compactness, approximation of functions in Lp by smooth functions. Using mollifiers is treated in a highly detailed fashion. Chapter 3 — Banach Spaces — complete normed vector spaces.
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This chapter introduces various examples, the lp spaces and the Lp spaces of functions. Convergence is discussed. Chapter 4 — Inner-Product Spaces and Hilbert Spaces — these spaces are very attractive to study since many of the properties of the Euclidean norm carry over to their norm. Amongst the material covered in this chapter are the Riesz representation theory and the spectral theory for compact self-adjoint operators.
Many consequences are discussed.
Chapter 6 — Linear Partial Differential Equations — consideration is limited to time independent problems. Applications include optimisation theory, linearised elasticity and linearised fluid mechanics. These are minimisation problems. Chapter 7 — Differential Calculus in Normed Vector Spaces — this is the start of nonlinear functional analysis, which focuses on the idea of derivability of mappings between arbitrary normed vector spaces.
A small excerpt can be seen here.
Nonlinear Functional Analysis | The Institute of Mathematical Sciences
Those with accounts on the Elsevier page can see the whole content. In essence, the field is nonlinear analysis. The paper and other papers by the same author contains terms like guage function, upper semicontinuous, ultrapower, nontrivial ultrafilter, etc. Looking up wikipedia for all these terms is not helpful at all. I do not get any sort of understanding apart from the basic definitions, which I soon forget.
Background: I have studied functional analysis mostly linear bounded functions, etc. I want to be able to understand this paper, without doing tons of more studying to be able to understand the arguments at least. Could someone kindly point me to set of online notes or something of the sort that would quickly get me acquainted with these terms and their applications, so that I could proceed with understanding the paper? Sign up to join this community. The best answers are voted up and rise to the top.