Mathematical Analysis: Linear and Metric Structures and Continuity
Springer Science & Business Media, Sep 4, 2007 - Mathematics - 466 pages
One of the fundamental ideas of mathematical analysis is the notion of a function; we use it to describe and study relationships among variable quantities in a system and transformations of a system. We have already discussed real functions of one real variable and a few examples of functions of several variables but there are many more examples of functions that the real world, physics, natural and social sciences, and mathematics have to offer: (a) not only do we associate numbers and points to points, but we as- ciate numbers or vectors to vectors, (b) in the calculus of variations and in mechanics one associates an - ergy or action to each curve y(t) connecting two points (a, y(a)) and (b,y(b)): b Lea ~(y) - / 9 F(t, y(t), y' (t))dt t. J a in terms of the so-called Lagrangian F(t, y, p), (c) in the theory of integral equations one maps a function into a new function b /1, d-r / o. J a by means of a kernel K(s, T), (d) in the theory of differential equations one considers transformations of a function x(t) into the new function t t f f( a where f(s, y) is given. 1 in M. Giaquinta, G. Modica, Mathematical Analysis. Functions of One Va- able, Birkh~user, Boston, 2003, which we shall refer to as [GM1] and in M. G- quinta, G. Modica, Mathematical Analysis. Approximation and Discrete Processes, Birkhs Boston, 2004, which we shall refer to as [GM2].
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algebra Banach space basis e1 bilinear form bounded called Cauchy sequence class C1 closed sets compact complete connected continuous function continuous map converges uniformly coordinates Corollary curve deﬁned Deﬁnition denoted dimension distance eigenvalues eigenvectors equation equivalent Euclidean Example exists ﬁeld Figure ﬁl ﬁnd ﬁnite ﬁnite-dimensional ﬁrst ﬁxed point Fourier function f hence Hermitian Hermitian product Hermitian space Hilbert space homeomorphism homotopy inequality infer inﬁnite injective inner product interval inverse isometry isomorphism Lemma Let f Let H linear map linear operator linear space linear subspace linearly independent Lipschitz Lipschitz-continuous map f matrix associated metric space Moreover nonzero normed space open set orthogonal projection orthonormal basis path-connected polynomials Proof Proposition prove respectively scalar self-adjoint operators Show space and let Span subset Suppose surjective Theorem trivially uniformly continuous unique solution vector space zero