1. Explain Call by Value and Call by Reference with appropriate…

Question Answered step-by-step 1. Explain Call by Value and Call by Reference with appropriate… 1. Explain Call by Value and Call by Reference with appropriate example.2. What is inline function? Explain with example3. Explain Function with Default Arguments with appropriate example4. What is friend function? Explain with example.5. Explain function overloading with example.1. (i) State the Closed Graph Theorem. Let (X, || · ||X) and (Y, || · ||Y ) be Banach spaces. (ii) Let T : X → Y be a linear and continuous operator such that Range(T) is closed. Prove that there exists C > 0 such that for every y ∈ T(X) there exists x ∈ X such that y = T(x) and ||x||X ≤ C||y||Y . (iii) Let T : X → Y be a linear operator such that for every sequence {xn} ⊂ X ||xn||X → 0 ⇒ T(xn) * 0 in σ(Y, Y 0 ). Prove that T is continuous, i.e., T ∈ L(X; Y ). 2. Prove that if (X, ||·||) is a normed space over R such that X0 is separable, then X is also separable. 3. (i) State and prove the Banach-Steinhaus Theorem for normed spaces. (ii) Let (X, || · ||X) be a Banach space over R and let Ln, L ∈ X0 , n ∈ N, be such that Ln ?* L. Prove that ||L||X0 ≤ lim inf n→∞ ||Ln||X0 < +∞. 4. (i) Give the definition of a compact operator between two normed spaces. (ii) Let (X, || · ||X) and (Y, || · ||Y ) be normed spaces, and let T : X → Y be a linear compact operator. Prove that T ? is also compact. 5. Let (X,(·, ·)) be a Hilbert space and let T : X → X be a linear, continuous operator such that (T x, x) ≥ 0 for all x ∈ X. (i) Prove that Ker(T) = (Range(T))⊥ . (ii) Prove that I + tT is bijective for all t > 0. [20 points.] Let `∞ be the space of bounded sequences of real numbers, equipped with the || · ||∞ norm, i.e., if x ∈ `∞ is the sequence x = (x1, x2, . . .), then ||x||∞ := sup n∈N |xn|. Let c be the set of all convergent sequences, and let c0 be the set of sequences in c with limit zero. Prove that c and c0 are closed subspaces of `∞ . 2. [20 points.] Consider R N = Πn∈NXn, where Xn = R for all n ∈ N, i.e., the space of all sequences x = (x1, x2, . . .), xn ∈ R for all n ∈ N. Let Y be the subset of R N consisting of all sequences that are “eventually zero”, that is, all sequences x = (x1, x2, . . .) such that xn 6= 0 for only finitely many n ∈ N. What is the closure of Y in the box and product topologies? 3. [20 points.] Let (X, τ ) be a completely regular topological space, and consider its Stone-Cech compactification ˇ β(X). Prove that X is connected if and only if β(X) is connected. 4. (i) [5 points.] State Baire’s Category Theorem. (ii) [15 points.] Let X be a complete metric space, and let F be a subset of C(X; R) such that for every x ∈ X, the set {f(x) : f ∈ F } is bounded. Prove that there is a nonempty open set U of X in which the functions in F are uniformly bounded, i.e., there exist M ∈ R such that supf∈F,x∈U |f(x)| = M. 5. (i) [5 points.] State Ascoli-Arzel`a Theorem. (ii) [15 points.] Let X be a compact metric space, and let fn ∈ C(X; R) be an equicontinuous and pointwise bounded sequence of functions. Prove that if every uniformly convergent subsequence has the same limit f ∈ C(X; R), then fn converge uniformly to f(15 points) Let X be an infinite set. Let τ = {U ⊆ X : XU is finite} ∪ {∅}. It is known that τ is a topology. (i) Show that if X is uncountable then τ is not first countable (i.e. does not satisfy the first axiom of countability). (ii) Let x ∈ X. Show that a sequence {xn}n=1,2,… converges to x if and only if for each n ∈ N either xn = x or there exists n0 ∈ N such that for all m > n0 xm 6= xn. (In other words no element except x can appear infinitely many times in the sequence.) 2. (25 points) We define a topology τ on R 2 as follows: subset U ⊂ R 2 beelongs to τ if at every point x ∈ U, U contains an open line segment through x in every direction, that is for every v ∈ S 1 there exists ε > 0 such that for every s ∈ (−ε, ε), x + sv ∈ U. (i) Prove that τ is a topology. What is the relation to the standard topology (weaker, stronger, neither)? What is the induced topology on any straight line of R 2 ? What is the induced topology on a circle? (ii) Prove that (R 2 , τ ) is separable and Hausdorff. (iii) Prove that there exists closed set E ⊂ R 2 which is equipotent to R and is such that the induced topology is discrete. Prove that (R 2 , τ ) is not normal. 3. (15 points) Let f : X → Y be a continuous and closed mapping. Assume that Y is compact and that for all y ∈ Y , f −1 ({y}) is compact. Show that X is compact. 4. (25 points) Let X = C([a, b], R) for some a < b. (i) Show that for p > 0, dp : X × X → R defined by dp(f, g) = max t∈[a,b] |f(t) − g(t)|e −pt is a metric on X. (ii) Show that for any p > 0 the metric dp generates the same topology on X as the standard metric on X: d(f, g) = max t∈[a,b] |f(t) − g(t)|. (iii) Let h ∈ X and K ∈ C([a, b] × [a, b], R). Show that there exists a unique f ∈ X which satisfies the equation f(t) = h(t) + Z t a K(t, s)f(s)ds for all t ∈ [a, b]. Hint: Use Banach contraction principle in (X, dp) for appropriately chosen p > 0. 5. (20 points) Consider the metric space X = C ([0, 1] , R) with the sup metric: d∞(f, g) = max x∈[0,1] |f(x) − g(x)|. For every n ∈ N let Xn := {f ∈ X : there is x ∈ [0, 1] such that |f (x) − f (y)| ≤ n |x − y| for all y ∈ [0, 1]} . (i) Fix n ∈ N and prove that each f ∈ X can be approximated by a zigzag (piecewise linear) function g ∈ X with sufficiently large slopes so that it does not belong to Xn and such that d∞ (f, g) is arbitrarily small. (ii) Fix n ∈ N and prove that every open set U ⊂ X contains an open set that does not intersect Xn. (iii) Prove that there exists a dense Gδ set in X that consists of nowhere differentiable functions. A set is a Gδ set if it is a countable intersection of open sets.6. Explain operator overloading with exampleWhat is inline function? Explain with example. Computer Science Engineering & Technology C++ Programming GIS 200 Share QuestionEmailCopy link Comments (0)