In this chapter we lay the theoretical foundations for the treatment of differential equations in Chapters 2 and 3. We begin in section 1. In section 1. The ideas and definitions of the theory are more elaborate than those of ordinary calculus; this is the price paid for developing a theory which is in many ways simpler as well as more comprehensive. In particular, the theorems about convergence and differentiation of series of generalised functions are simpler than in ordinary analysis. This is illustrated by examples in sections 1.
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In this chapter we lay the theoretical foundations for the treatment of differential equations in Chapters 2 and 3.
We begin in section 1. In section 1. The ideas and definitions of the theory are more elaborate than those of ordinary calculus; this is the price paid for developing a theory which is in many ways simpler as well as more comprehensive. In particular, the theorems about convergence and differentiation of series of generalised functions are simpler than in ordinary analysis. This is illustrated by examples in sections 1. References to other accounts of this subject are given in section 1.
The theory of generalised functions was invented in order to give a solid theoretical foundation to the delta function, which had been introduced by Dirac as a technical device in the mathematical formulation of quantum mechanics. But the idea of Dirac's delta function can easily be understood in classical terms, as follows. Consider a rod of nonuniform thickness. But if the mass is concentrated at a finite number of points instead of being distributed continuously, then the above description breaks down.
Suppose that the bead has unit mass and is so small that it is reasonable to represent it mathematically as a point. Then the total mass in the interval a,b is zero if 0 is outside the interval, and is one if zero is inside the interval. There is no function p that can represent this mass-distribution.
But if a function vanishes everywhere except at a single point, it is easy to prove that its integral over any interval must be zero, so that integrating it over an interval including the origin cannot give the correct value, 1. This makes good physical sense, though it is mathematically absurd. It can be considered as a technical trick, or short cut, for obtaining results for discrete point particles from the continuous theory, results which can always be verified if desired by working out the discrete case from first principles.
A point particle can be considered as the limit of a sequence of continuous distributions which become more and more concentrated. The delta function can similarly be considered as the limit of a sequence of ordinary functions.
Consider, for example,. The delta function can be considered as a kind of mathematical shorthand representing that procedure, and results obtained by its use can always be verified if desired by working with dn and then evaluating the limit.
The point of view described above is that of many physicists, engineers, and applied mathematicians who use the delta function. To the pure mathematicians of Dirac's generation it presented a challenge: an idea which is mathematically absurd, but still works, and gives useful and correct results, must be somehow essentially right. The situation is reminiscent of the use of complex numbers in the 16th century for solving algebraic equations.
It proved useful to pretend that -1 has a square root, even though it clearly has not, since one could then use an algorithm involving imaginary numbers for obtaining real roots of cubic equations; any result obtained this way could be verified by directly substituting it in the equation and showing that it really was a root. It was only much later that complex numbers were given a solid mathematical foundation, and then with the development of the theory of functions of a complex variable their applications far transcended the simple algebra which led to their introduction.
The solid foundation was developed by Sobolev in and Schwartz in the s , and again goes far beyond merely propping up the delta function. The theory of generalised functions that they developed can be used to replace ordinary analysis, and is in many ways simpler.
Every generalised function is differentiable, for example; and one can differentiate and integrate series term by term without worrying about uniform convergence.
The theory also has limitations: that is, it shows clearly what you cannot do with the delta function as well as what you can — namely, you cannot multiply it by itself, or by a discontinuous function. The other disadvantage of the theory is that it involves a certain amount of formal machinery. However, we must be careful about what functions we allow as weighting functions. The definitions below may at first seem arbitrary and needlessly complicated; but they are carefully framed, as you will see, to make the resulting theory as simple as possible.
The reader unfamiliar with the notation of set theory should consult Appendix A. Definition 1. A function has bounded support if there are numbers a,b such that sup f [subset] [ a,b ].
The set of all test functions is called P. This is probably the simplest example of a test function. They are bound to have somewhat complicated forms, for the following reason. Test functions are thus peculiar functions; they are smooth, yet Taylor expansions are not valid. Fortunately, we never really need explicit formulas for test functions. They are used for theoretical purposes only, and are well-behaved smooth etc.
The following result gives another nice mathematical property. Proposition 1. A set of functions with this property is often called a 'space', for reasons that will become clear in Chapter 4. Examples 1. It is not linear. We must now define convergence in the space P. The reader will know that more than one meaning can be attached to the phrase 'a sequence of functions is convergent'.
For some purposes pointwise convergence is suitable; for other purposes uniform convergence is needed an outline of the theory of uniform convergence is given in Appendix B. One of the characteristics of functional analysis is its use of many different kinds of convergence, as demanded by different problems.
The most useful for our present purpose is the following. This is a stringent definition, much stronger than ordinary convergence. We do not offer an example because specific examples are never needed: test functions are only the scaffolding upon which the main part of the theory is built. A continuous linear functional on P is called a distribution , or generalised function. This can be shown to be equivalent to the condition that f map every convergent sequence of numbers into a convergent sequence.
We adopt the latter as our definition in P, because there is no analogue in P of the modulus of a number which appears in the other definition in Chapter 4 we shall consider this question further. Notation 1.
Applied Functional Analysis
This book is intended to be a simple and easy introduction to the subject. It is aimed at undergraduate students of mathematics, and mathematical physics and engineering, though I hope to interest other readers too. I have tried to avoid difficult ideas, as far as possible; the spectral theorem, for example, is discussed for operators with discrete eigenvalues only. But within this limitation I have tried to tell a coherent story, and to give both proofs and motivation for the theory, at least in the central Parts II and III; in Parts I and IV some proofs are omitted.
Would you like to tell us about a lower price? If you are a seller for this product, would you like to suggest updates through seller support? A stimulating introductory text, this volume examines many important applications of functional analysis to mechanics, fluid mechanics, diffusive growth, and approximation. Detailed enough to impart a thorough understanding, the text is also sufficiently straightforward for those unfamiliar with abstract analysis.