(b) Prove that the set S = {x EQ: √2

Q: Prove that If f is holomorphic on D(0, 2), f(0) = -i and |ƒ(z)| > 1 on the unit circle then f has at…

A: It is given that f is holomorphic in the disc with centre 0 and radius 2. So, the function f is…

Q: et [a, b] be a real interva b]→X is continuous, the is continuously differentia

A:

Q: H is a Hilbert space, A: H → H is linear ar = for all r, y e H. Use th closed graph theorem to show…

A: For two normed spaces V and W, a linear operator T:V→W is closed if for each xn∈V, xn→x and Txn→y…

Q: 3. Let X and Y be normed spaces and f:X→Y be a bijection linear function. prove or disprove f is…

A: It is true that if f is a bijection linear function from a normed space X to a normed space Y, then…

Q: Let X be a normed space and let 0x0 EX. Show that there is f = X* such that f(x) = 1 and ||f|| =…

A: Given, a normed space  and .It is required to show that, there exists  (the dual space of ) such…

Q: Let (X1, d1) and (X2, d2) be separable metric spaces. Prove that product X1 × X2 with metric…

A: A matric space X is separable if it has a countable dense subset.

Q: Consider R2 equipped with the Euclidean metric d2. Let A be a non-empty subset of R2 , that is…

A: Consider R2 equipped with the Euclidean metric d2. Let A be a non-empty subset of R2 , that is…

Q: (b) Show that the set A = {(x, y) E R² : y > 0} is not complete.

A: Given: The given set ,

Q: Q. No. 2 Let N be the set of natural numbers and N = N U {∞}. Define a function d: N* × N* → R as…

A: I have proved the three conditions for metric

Q: Let (X, d) be a metric space. Suppose that (xn) is a sequence of points in X which converges to a…

A:

Q: AP1) Prove that f: E → R; x x² is uniformly continuous if E is bounded. AP2) Prove that f : R R; x x…

A: Definition: A function fx said to be uniformly continuous, if for every ε>0, there exists δ>0…

Q: 3.5 Show that the metric space (R, d), where d(r, y) = |x – y|, is separable.

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: (b) Give an example of a bounded function f on [a,b] for which f2 e R[a,b), but f is not in R[a,b).

A:

Q: (d) Show that T is bounded and find ||T||. (e) Find the adjoint T* of T.

A: We will use the vector/matrix notations to visualise things easily.

Q: Give an example each of the following. Justify your choice of examples.  i) A non-trivial set in…

A: We have to give an example for each of the following statements ; (i) A non trivial set in Euclidean…

Q: Suppose A is a function that assigns to each subset of the plane a number in such a way that the…

A: Given that AX≥0 and if X and Y are disjoints then AX∪Y=AX+AY To prove if X⊂Y then AX≤AY It is known…

Q: E. Let H be the set of all points (x, y) in R² such that x² + 3y² = 12. Show that H is a closed…

A: We have to find H is closed  subset bounded.

Q: (K", || |,) is called the LP-norm on K". is a normed linear space for 1 <PS00. || |,

A:

Q: Let B = {(p,q) e R² : p² + q² = 1}. (a) Show that B is NOT disconnected (b) Given that there is no…

A: Given: B=p,q∈ℝ2 : p2+q2=1. There is no continuous mapping f:B→ℝ. To prove: B is not disconnected.…

Q: 1. Let V be an inner product space. Suppose f,g E V such that f -gg (a) Prove that 1I2 = |lf- gll2 +…

A: To prove the required norm property in an inner product space, under the given conditions

Q: Q2:- Let (X, d) be a metric space. Xo, Yo EX and r, s > 0. Show that B, (yo) Br(xo) if s+d(xo, yo)…

A:

Q: Let X= [0, 0 ), and defined d as d(x,y)= e(xy) then verify all four condition of metric space

A:

Q: 2.1 Let B be the set of all bounded sequences of real numbers and define the function d: B x B R by…

A:

Q: Let 7,V be a projection af the Vect or space. Accordling to this: Show that every Xe Im (T) for…

A: The given problem is related with projection. Given that T is a projection of the vector space V .…

Q: Prove that A = {x : x € X and ƒÃ(x) = 0}. (iii) Suppose A and B are nonempty disjoint closed subsets…

A:

Q: 3.3 Let r1 and r2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

A:

Q: 3. Show that Z[√2i] is a Euclidean Domain with the function N(z) = |zz|.

A: Given function Nz=zz¯ We need to show that ℤ2i is a Euclidean Domain with the function. A Euclidean…

Q: If f is a continuous function satisfying f(x) = x* - x+(1-x)[,f'f)dt + x[ (t -r°)f(@dt for all x,…

A:

Q: Let (X, A, µ) and (Y, B, v) be measure spaces. Prove that (a) if E E A x B then y-section of E is…

A:

Q: sup norm

A:

Q: 4. Let C₁ be the set of all continuous real-valued functions on I = [0, 1]. (a) Show that d : C₂ ×…

A: According to our guidelines I can answer only Question no. 4(a).

Q: (iii) (f/g)(x) = g(x) if g(x) #0 for all z € M. Prove that cf. fg and (f/g) are all continuous. 8.…

A: To Prove: Restriction of a continuous function is also continuous on Metric space.

Q: 5)Evaluate F - nds for F = y²zi + y°j+ xzk using Gauss' Theorem. Consider that S is the boundary of…

A: We will find out the required value.

Q: Let (X, d) be a metric space. The diameter of a non-empty subset M of X is defined as 8(M)= sup d(x,…

A:

Q: metric

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Q: 2. Let U = {a + bx + cx2 + d³ e P3 (R) | a- 6+d3D0}, and let W = {a + bx + ca? + dx³ e P3(R)| a + 2c…

A:

Q: Let A C Rª be a convex set. f : A → R is a strictly quasi-concave function if and only if for any a…

A:

Q: Let U be a family of real-valued L-Lipschitz functions on some metric space X. Suppose that there is…

A: Let U be a family of real-valued L-Lipschitz functions on some metric space X.Suppose that there is…

Q: Let (X, d₁) and (Y, d₁) be metric spaces. Let f: X Y be continuous function, then f-¹(G) is open in…

A: Solution

Q: Suppose X is a topological space, Y is a Hausdorff space, and f:X→Y is continuous. Show that the…

A: Suppose X is a topological space, Y is a Hausdorff space, and f:X→Y is continuous. Show that the…

Q: Suppose that f1 : [0, 1] → R and f2 : [0, 1] → R are continuous everywhere and that f1(0) f2(1).…

A: Intermediate value theorem : Suppose be a continuous function then it takes all values between .

Q: Let E be a Euclidean space with inner ratic form. For all u, v E E, we have the C y(u, v)² ≤…

A: