TABLE OF CONTENTS In the previous chapter, it was suggested that the infinite dimensional analysis is necessary to make the calculus of variation in a rigorous way. The key word is the completeness and this chapter is devoted to it. Thus, we shall describe the theory of Hilbert spaces, Fourier series, and eigenvalue problems.
Remember that norm induces metric in the vector space, which is said to be a Banach space if it is complete with respect to that metric. Let ( L, ? ?) be a Banach space and T : L ? R be a linear mapping. Sometimes T is referred to as an operator. It is said to be bounded if there is a constant M > 0 satisfying
for any f ? L.
This is equivalent to saying that T is continuous at any or some element in L because of its linearity. In fact, if (3.1) holds and f n ? f in L, then it follows that
Therefore, T(f n) ? T(f) follows. Conversely, if T is continuous at f = 0, then it is bounded. In fact, if this is not the case, there is a sequence { f n} ? L such that
for n = 1, 2, . Because T is linear, it holds that T (0) = 0 and hence f n ?
