If equipped with an additional operation of multiplication of vectors by other vectors (to give yet other vectors), an operation that is well-intertwined with the existing linear and topological structures, a Banach space becomes a Banach algebra. Such an additional operation is naturally defined in the space of continuous functions by pointwise multiplication. In the space of integrable functions on the positive half-axis, however, the role of multiplication is most naturally played by convolution. We use this additional algebraic structure in the discussed space to study the McKendrick–von Foerster model of population dynamics. The existence and uniqueness of the renewal equation that is a key to the model turns out – surprise, surprise! – to be the result of the completeness of the underlying space.

Bounded linear operators form a natural class of maps between normed linear spaces: they are, by definition, linear and continuous, so that, in other words, they preserve, at least to some extent, the linear and the topological structures of the normed spaces involved. As it turns out, bounded linear operators from one normed linear space to another form a normed linear space themselves. Moreover, if the range space is complete, the space of operators is complete also, and is thus a Banach space. We note that we have already encountered examples of bounded linear operators on the previous pages of this book and discuss a score of new ones. Nor do we refrain from calculating norms of some of them.

We prove two versions of the celebrated theorem of Picard: the local and the global existence and uniqueness results for differential equations. We use Banach’s principle as a main tool in our analysis, and this makes us realize that it is the completeness of the space of continuous functions that is the reason for the existence of solutions to differential equations. In the mean time we get acquainted with the notion of equivalent metrics and learn that, in proving the existence of a fixed point of a map, it is sometimes more convenient to use one norm and sometimes another, equivalent one.

A sequence of norm-one elements of a Hilbert space that are mutually orthogonal is said to form an orthonormal sequence. If, additionally, such a sequence spans the entire space, it is said to be complete. As it turns out, if in a Hilbert space there is a complete orthonormal sequence, this space is indistinguishable from the space of square summable sequences. In particular, perhaps contrary to our misleading intuition saying that there are many more square integrable functions than there are square summable sequences, the space of the former is as large as (in fact much the same as) the space of the latter. We will see one important consequence of this stunning result in the next chapter.

The Weiestrass theorem says that any continuous function on a finite closed interval can be uniformly approximated, with any required accuracy, by polynomials. The Stone–Weierstrass theorem extends this result to an abstract setting, where the interval is replaced by a compact topological space, and the role of polynomials is played by a class of functions that enjoy certain properties mimicking those of polynomials. There are scores of proofs of the latter result; the one presented in this little book could not fail to stress the importance of completeness of the space of continuous functions.

The space of bounded linear operators mapping a Banach space X into itself is not only a Banach space but also a Banach algebra with multiplication defined as composition. This provides additional possibilities of manipulation with elements of the space of operators. In particular, we can use `power series’ of operators to construct inverses of other operators, and thus solve linear equations in X. We can also define exponential functions of bounded linear operators to solve differential equations in X. Again, all of this would be impossible, were we not working in a complete space.

We are finally introduced to the fundamental notion of functional analysis: the Banach space, a unique blend of notions of linear algebra and metric topology. We get to know a number of classical, elementary Banach spaces. Also, examples of normed linear spaces that are not complete teach us that in a Banach space its `extent’ and its norm match each other tightly.

The chapter is a gentle introduction to the theory of strongly continuous semigroups of operators. We present the notion of the generator, discuss the generator’s basic properties and study a number of examples. We learn that the way to discover whether a given operator is a semigroup generator is by examining the resolvent equation, and are thus naturally led to the Hille–Yosida–Feller–Phillips–Miyadera theorem that characterizes generators in terms of resolvents. Two valuable consequences, the generation theorems for maximal dissipative operators in Hilbert space and operators satisfying the positive-maximum principle in the space of continuous function, are also explained. This material is supplemented with three theorems on the generation of positive semigroups. The reader of this book, however, will undoubtedly have noticed that the whole theory would have failed were it not for the fact that we are working in Banach spaces; without the assumption of completeness, we could not be sure that the Yosida approximation converges, and the entire reasoning would have collapsed.

The fact that the space of square integrable functions on a finite interval is quite the same as the space of square integrable sequences provides a way to solve the heat equation, one of the fundamental equations of mathematical physics (and of the theory of stochastic processes). As originally posed in the former space, the equation seems to be rather difficult. But the isomorphism between these spaces transforms the equation into a series of ordinary differential equations with constant coefficients, and these can be solved explicitly. On the level of calculations, we are simply using the well-known method of separation of variables of the theory of partial differential equations; more intrinsically, however, we are looking at the method from a proper perspective, the perspective of Hilbert spaces.