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1 that in an infinite-dimensional Banach space X, a sequence {xn}REIN C {x E XI IlxlI < 1} and an e > 0 always exist, such that Ilx. - x,,, 1I > e, Vn # m. In the special case of a Hilbert space X, for example, for every orthonormal system one has Ijx - xmII = v2, Vn 54 m. Luckily, a topology on a Banach space is available which (a) is coarser than the norm topology on X, and (b) in the sense of the comment (ii) above, has sufficiently many compact sets. 2 Applying the Fundamental Theorem in Banach Spaces 21 elements f of the dual space X' of X are continuous.

Oo always implies that f(x) -. +oo. 4. The coerciveness condition will often be made more stringent in later applications to take into account a specification of the growth of f(x) for lixil -. oo. Furthermore, we shall discuss a few sufficient conditions for coerciveness. It is worth emphasising that very many solutions to minimisation problems can be achieved via a realisation of the coerciveness condition. If f : M -. 1R, is a coercive function, then all the sets Mf,a, A E IR are bounded. If X is now a reflexive Banach space, which is fortunately the case in most applications, then the sets Mf,a, A E IR are weakly relatively compact, due to Eberlein's Theorem.

E. f : E, -, 1R, then bsa f is an element of the dual space Ei. We have already introduced the concept of the Frechet derivative Deaf. 3 The Gateaux Derivative 47 of all partial derivatives. The next lemmas show the corresponding relationship in the general case between Gateaux and Frechet derivatives. 2. Let f : El -* E2. If the Gateaux derivative 6x0 f exists in a neighborhood U of xo and is continuous, then Proof. We put w(xo,h) = f(xo + h) - f(xo) - 6xof(h). There exists for all y' C E2 a T E [0,1] such that (w(xo, h), y') _((6x0+rh(f) - 6xo (f))(h), y').

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The integrals of mechanics by Lester O.

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