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This section is rather brief since we suppose that the reader already has some knowledge of linear algebra. Therefore, it should be viewed mainly as a source of concepts and notation which will be frequently used in the sequel. There are plenty of textbooks on this subject. As we are interested in applications to analysis we recommend to the reader the classical book Halmos [64] to find more detailed information.

In this section we point out some fundamental properties of linear operators in Banach spaces. The key assertions presented are the Uniform Boundedness Principle, the Banach-Steinhaus Theorem, the Open Mapping Theorem, the Hahn-Banach Theorem, the Separation Theorem, the Eberlain-Smulyan Theorem and the Banach Theorem. We recall that the collection of all continuous linear operators from a normed linear space into a normed linear space is denoted by , and is a normed linear space with the norm .

This short section is devoted to the integration of mappings which take values in a Banach space . We will consider two types of domains of such mappings: either compact intervals or measurable spaces. For scalar functions the former case leads to the Riemann integral and the latter to the Lebesgue integral with respect to a measure.

In this section we are looking for conditions which allow us to invert a map : → , especially : ℝ → ℝ. The simple case of a linear operator indicates that a reasonable assumption is that = .

One of the most frequent problems in analysis, especially in its applications, consists in solving the equation () = where is a mapping from a Banach space into a Banach space . Such an equation can be reduced to the equation () = , or, provided ⊂ , to the equation () = . (5.1.1) In this section we present two basic results on the solvability of (5.1.1) in a special case, namely, for a continuous mapping and a finite dimensional , and a compact mapping in a general Banach space of infinite dimension — the Brouwer and the Schauder Fixed Point Theorems.

In this section we present necessary and/or sufficient conditions for local extrema of real functionals. The most famous ones are the Euler and Lagrange necessary conditions and the Lagrange sufficient condition. We also present the brachistochrone problem, one of the oldest problems in the calculus of variations. We also discuss regularity of the point of a local extremum. The methods presented in this section are motivated by the equation () = 0 (6.1.1) where is a continuous real function defined in ℝ. The solution of this equation can be transformed to the problem of finding a local extremum of the integral of (i.e., () = (), ∈ ℝ). Indeed, if there exists a point ∈ ℝ at which the function has its local extremum, then the derivative () necessarily vanishes due to a familiar theorem of the first-semester calculus. The problem of finding solutions of (6.1.1) can be thus transformed to the problem of finding local extrema of the function . On the other hand, one should keep in mind that the equation (6.1.1) may have a solution which is not a local extremum of .

In this section we will explain the notion of the classical solution of a semilinear problem with the Laplace operator and explain what is the “right” functional setting for it. Let Ω be an open bounded subset of ℝ and let : Ω →; ℝ be a real smooth function. We will denote by the defined in Ω. Let be a continuous real function. We will study the Dirichlet boundary value problem and look for its . Following the definition of the classical solution for the ordinary differential equation it should be a function such that () = 0 for every ∈ ∂Ω and the equation −Δ() = (()) is satisfied at every point ∈ Ω. Let us explain why this is not a suitable definition of the solution for partial differential equations.