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.

Guided by basic intuitions, we introduce the notion of a complete metric space and discover that we have in fact encountered it before in our study of mathematics. In particular, we learn that if the set of real numbers were not complete, bounded increasing (or decreasing) sequences would not have limits. Similarly, we realize that if time were not complete, Achilles would never catch the tortoise. In a slightly more advanced part, we show that criteria for convergence of functional series involve the notion of completeness of the space of continuous functions.

Although there is a particular beauty in the statement that the space of bounded linear operators in a Banach space is itself a Banach space, the norm in this space is more often than not too strong to encompass more delicate convergence theorems of contemporary mathematics. Strong convergence is a notion that is more suitable for such purposes. We exemplify this by studying two classical theorems: Bernstein’s approximation of continuous functions by polynomials and the theorem of Fej\’er on convergence of Fourier series. In both cases the operators involved converge strongly but not in the operator norm. Before doing that, however, we discuss the theorem of Banach and Steinhaus. This result ensures in particular that in Banach spaces strong convergence of bounded linear operators implies boundedness of the limit operator. The chapter also covers the famous Poisson approximation to the binomial, the only example of a limit theorem of probability known to the author that can in fact be stated in the framework of norm convergence of operators.

The Riemann integral for vector-valued functions can be defined in the same way as for scalar-valued functions. Moreover, the theory of the so-defined integral is rather similar to the classical one. In particular, any continuous function with values in a Banach space is Riemann integrable, and the fundamental theorem of calculus remains valid. The theory of Riemann integration for functions with values in a normed, not complete, space would not be so elegant.

A Hilbert space is specific example of a Banach space, because its norm comes from a scalar product. This particular norm makes the geometry of a Hilbert space very familiar to us. In particular, in a Hilbert space one can find a unique element of a closed, convex subset that minimizes the distance of this subset from a point lying outside of it. One can also think of projections of vectors on closed subspaces. Again, all this would have been impossible were the space with scalar product not complete. The chapter ends with remarkable example showing that conditional probability, one of the fundamental notions of probability theory, has much to do with projections in a Hilbert space.

Banach’s principle states that if a map T uniformly reduces the distance between points of a complete metric space, then there is a unique x such that Tx = x, called T’s fixed point. This simple statement has profound and surprising consequences, as we will see in the following chapters. For now, we will content ourselves with an example, which may appear to belong to the realm of linear algebra, but is, in fact, much easier to deal with using metric notions.