Let X= [0,100] x [-2,2] with the Euclidean metric d on X and T: X→→ X be defined by T(a,b) = (2+ √a² - 8a+ 40, tan™ -1/2)0 for all (a, b) e X. Prove that T is a Banach contraction mapping with the metric d.
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- Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are linearly dependent in the vector space C[0,1], but linearly independent in C[1,1].if (X,d) is discrete metric space, ACX and A+: (х,d) Find the sets of A° and A (functional analysis )Q5. Describe all linear maps f : R → R when R is regarded as an R-vector space
- Show that ℝ2 = span([1 1], [1 -1])(3) Let R² and R be endowed with their Euclidean metrics. (a) Show that projections p₁(x, y) = x and p₂(x, y) = y are continuous. (b) Show that the direct image of an open set under p₁ or på is open.3. This question concerns the L metric d and the L' metric e on C[0, 1] (Examples 1.1 1)). a) Calculate d(S,9) and e(f,g), where f(z) = 1+ z and 9(z) =z. b) Explain why we have e(f.g) s d(f.g) for all functions f and g. Under what conditions on f and g is e(f.g) = d(f,g)? %3D In the remainder of the question, we treat an explicit example of the "function with a very narrow tall bump" of Figure 1.14. For each integer n 2 2, define a function InE C10,1] by Tz+1- Sn(z) -nz +1 + if ss+4 otherwise. e) Sketch the graph of fs, and describe the graph of fn for any n. d) Let g(x) 0 be the zero function. Calculate d(fn.g) and e(fn.g). e) ** Deduce that there is no number A such that elf.9) Sd.g) S Ae(f.g) for all f.ge C0,1.
- Q3:- Let ((0,4], d₁) be a metric space and the open ball in this metric B₂ (3) = (1,4]. Show that (0,1] is closed set in this meric.Question #1 (a). Define three different metrices dı, d2, dz and their induced norms on R³. Also prove the equivalence of these norms. (b). Let X be set of all continuous real-valued functions on [a, b). Define d and d such that (X, d) is complete and (X, d) is not complete.(d) Passing through the point (0, 1, 2), perpendicular to the line L and intersects L. L(xo, v) = TER
- (a) Let M R. Give the radius r and the center c of B(-2,5) n B(6, 7). (b) Let M = R. Using interval notations, give a simplified expression for the set L= [B(1,1) UB(5,1)] n B(3,2). (c) Let = 3(-1)" + and A = {,: ne N). Give the set W of accumulation points of A. (d) Let A₁-(-2n, 2n) and K = U An. Give a simplified expression for the interior Kº of K. NEN (e) Give the set T of isolated points of B(0, 1) U (3) U (2,5,7) (f) Let [P,Q] be a segment in R2 with midpoint H, and let (1,0) and (2, 1) be the components of the points P and H, respectively. Give the components (x, y) of the point Q. (g) Suppose that f: (M, d) →R satisfies d(f(x), f(y)) - vl. Let 21 e M and set n+1 = f(n), for n e N". Assume that M is complete and let a = lim zn. Give a simple formula satisfied by a.Show that (C(X),|| · ||∞,x) is a metric space.5. Show the span{(1,0, 1), (-1,2, 3), (0, 1, – 1)} is all of R'.