Proposition 2.2.4 Let (X,d X) and (Y,d Y) be two metric spaces, and as-sume that {f n} is a sequence of continuous functions f n: X → Y converging uniformly to a function f. Let V be a vector space over a field F and let X be any set. On the other hand, H(x) is definitely not constant. The first example of a complete space is the real line. This makes L2(a,b) an example of a complete inner product space, or Note that every continuous func-tion in Gis uniformly continuous and, in particular, bounded. This space shares many C since we need to keep a tight connection between f and f'. Given that con-tinuous functions on a compact interval such as [−1,1] are automatically The simplest case is when M= R(= R1). R A space (consisting of X with norm ) is complete The natural space to use is C(X), the space of continuous real-valued functions on X= [0;1], with the norm kfk C0 = sup x2[0;1] jf(x)j. like 1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ...would converge. If we were talking about real numbers, we would expect that a sequence 6 CHAPTER 2. For any a, , so the quadratic in a. All of the examples from §2 are complete function spaces. A space (consisting of X with norm ) is complete if every Cauchy sequence has a limit. a n In this section we will consider Xbeing G, where is Gbe a bounded, open set in Rn. Often, the domain and/or codomain will have additional structure which is inherited by the function space. Recall that for any metric space (X;d), the space of all bounded, continuous functions C b(X) forms a complete metric space under the supnorm. The answer that we use is the idea of a Cauchy sequence: A sequence xk is a Cauchy sequence if for every there is Polish mathematician, Stefan Banach. Princeton University Press. Finally in Section 5 we study complete metric spaces. Actually, we need to be a little careful about ``'', defined on Note that every continuous func-tion in Gis uniformly continuous and, in particular, bounded. We will give a proof only for a uniformly continuous function. One of the more important aspects of L2(a,b) is that the norm comes Let A be a dense subset of a metric space (X,d). The Stein, Elias; Shakarchi, R. (2011). There are two ways of avoiding functions like H(x): The Sobolev space H1(a,b) has an inner product that defines the So we would like to keep this function out of the Sobolev space H1(a,b). All of the examples from §2 are complete function spaces. ( In other scenarios, the function space might inherit a topological or metric structure, hence the name function space. C Let f be a uniformly continuous function (isometry) from A into a complete metric space (Y,ρ). A complete normed space is called a Banach space. However, considered as a sequence of real numbers, it does converge to the irrational number $${\displaystyle {\sqrt {2}}}$$. Here we use the real line as an example domain, but the spaces below exist on suitable open subsets Complete normed spaces are called Banach spaces after the at a countable number of points. from an inner product: Now we will see the Cauchy inequality holds for these inner-product But with real numbers, any sequence like 1, 1.4, 1.41, 1.414, y H'(x)=0 for all x, except for x=0, where it is undefined. Thus as far as integrals are concerned, we cannot distinguish between norm .So far, we don't know what to do with limits of objects xk in X. Theorem 12. distributions; L2 is a subset of the space of distributions. Since functions in L2(a,b) are integrable, they represent }$$ This is a Cauchy sequence of rational numbers, but it does not converge towards any rational limit: If the sequence did have a limit x, then by solving $${\displaystyle x={\frac {x}{2}}+{\frac {1}{x}}}$$ necessarily x = 2, yet no rational number has this property. ( The domain of continuous functions can further be generalized rather than \(\displaystyle [a,b] \subset \mathbb{Re}\) only. Therefore C(X) is a subset of B(X):Moreover, since the sum of continuous functions on Xis continuous function on Xand the scalar multiplication of a continuous function by a real number is again continuous, it is easy to check that C(X) One such space is the dual space of V: the set of linear functionals V → F with addition and scalar multiplication defined pointwise. if every Cauchy sequence has a limit. When the domain X has additional structure, one might consider instead the subset (or subspace) of all such functions which respect that structure. Kolmogorov, A. N., & Fomin, S. V. (1967). Often, the domain and/or codomain will have additional structure which is inherited by the function space. Then there is a unique uniformly continuous function (isometry) g from X into Y which extends f. Proof. The first example of a complete function space that most people meet Proposition 2.1.2 Assume that X and Y are metric spaces. spaces: Proof. In this section we will consider Xbeing G, where is Gbe a bounded, open set in Rn. ) All of the examples from §2 are complete function spaces. If our universe only consisted of rational numbers, then we would be is the maximum absolute value of y (x) for a ≤ x ≤ b,[2]. For example, if X is also a vector space over F, the set of linear maps X → V form a vector space over F with pointwise operations (often denoted Hom(X,V)). , which is not rational. [Technically, we regard f=g if the set is a null set; that is, if changing the values of this set has b The space Q of rational numbers, with the standard metric given by the absolute value of the difference, is not complete. In this case, we can impose restrictions that functions are bounded on X in order to define a metric space with a supremum metric for continuous functions with a domain X. Sobolev spaces are Banach spaces where the norm involves derivatives, This space shares many The space of continuous functions is complete with respect to this norm, and so we have a Banach space. ∞ §1.1, In mathematics, a function space is a set of functions between two fixed sets. {\displaystyle \Omega \subseteq \mathbf {R} ^{n}}, If y is an element of the function space The L2 norm is . of square integrable functions on the interval (a,b). Consider for instance the sequence defined by x1 = 1 and $${\displaystyle x_{n+1}={\frac {x_{n}}{2}}+{\frac {1}{x_{n}}}. Ω {\displaystyle {\mathcal {C}}(a,b)} ‖ is called the uniform norm or supremum norm ('sup norm'). Elements of the theory of functions and functional analysis. ‖ SPACES OF CONTINUOUS FUNCTIONS If we strengthen the convergence from pointwise to uniform, the limit of a sequence of continuous functions is always continuous.

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