The set S = {T, (U) | U is open in X} U {T, (V ) | V is open in Y } is a subbasis for the product topology on X x Y.

Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y.

A:

Q: Let z=x+iy ∈ C where x∈R\{1}, y∈R, and |z|=1. If w = (z+1)/(z-1), then show that Im(w) = y/(x-1).

A: Use the following formulae, to obtain the required result. Real part of a+ib=aImaginary part of…

Q: Let (X,T) and (Y,T1) be two topological spaces and let f be a continuous mapping of X into Y. *

A:

Q: f(x,y) = (In(y + x²)) x² + y² D = {(x,y) ERIy2x²-1 and y ≤4-e-*) (b) Observe the set Is D…

A: (b) The given set is D=x,y∈ℝ : y≥x2-1 and y≤4-e-x To find: The given set is bounded or closed or…

Q: Bounded by the planes z= x, y =x, x+y=2 and z= 0

A:

Q: Find the image of the semi-infinite strip x ≥ 0, 0 ≤ y ≤ π under the transformation w = exp z, and…

A: w lies in the portion of the closed upper half-plane external to the open unit disk.

Q: f(x,y) = 4xy² – x²y² – xy³

A: Given f(x,y)=4xy2-x2y2-xy3 and a triangular  region D with vertices (0,0) ,(6,0) and (0,6) Case 1 To…

Q: Prove the following theorem: If ƒ(x, y) is defined in an open region R of the xy-plane and if ƒx and…

A:

Q: (c) If X= R, with r={ R, ¢, Q, Irr}. Show that (R, t) is a compact space.

A:

Q: Let A = { (x,y)e RXR; 0<max{}x-31. ly-21)<1}, then: %3D None of the choices OA is not compact…

A:

Q: If h:(R,+) → (R*,.) is a mapping defined by h(x)=10*, VXER, then (R,+) = (R*,.).

A:

Q: Consider P2 with the inner product (p, q) = | | p(x)q(x) dx . Let U = span{1, 2x + 2} and g(x) = 3x2…

A: See the detailed solution below.

Q: 4. Consider the space (R,T), where T={R,Ø}u{AcR:[0,1)cA}. If A=(0,1), then the closure of A is: * O…

A:

Q: Let f(x) = 21 - x2 and g(x) = x2 + 3. Use symmetry, if appropriate, to help find the center of…

A: Consider the given two function f(x) and g(x).

Q: {{x, z}, {x}, {y}}. Let T be the topology generated by B and Let X = {x, y, z} and B = consider A =…

A:

Q: 4. Consider the space (R,T), where T={R,Ø}u{AcR:[0,1)cA}. If A=(0,1), then the closure of A is:…

A: consider the space (IR,T)where, T = IR,ϕ∪A⊂IR[0,1]⊂AIf A = (0,1), then closure of A is

Q: View S' as the boundary of the 2-disk D² = {(x, y) E R² | x² + y? < 1}. Fix To E S', and let X = (S'…

A: Given : D2=x,y∈ℝ2|x2+y2≤1 where x0 belongs to S1 which the boundary of 2-disk. X=(S1×S1)∪(D2×x0) To…

Q: Let (X, T1) and (Y, T2) be topological spaces. Show that the set {U × Y ∣ U ∈ T1} ∪ {X × V ∣ V ∈…

A:

Q: Let A = Z. Let R be defined on A x A where a Ry if 4 (x- y). Prove that R is transitive.

A:

Q: 28. Bounded by the planes z = x, y = x, x + y = 2, and z

A: Solution:-

Q: Let F(x, y, z) = (x-1z, y¬lz, In(xy). (a) Verify that F = V ƒ, where ƒ(x, y, z) = z In(xy). (b)…

A: Given Fx, y, z=x-1z, y-1z, lnxy (a) Verify for F=∇f where fx, y, z=z lnxy The gradient of a function…

Q: 4. Let {Xn, dn fnɛN be a collection of metrič spaces of D(x, y) = sup{d,(xn, Yn)/n | n E N} a metric…

A: Here we show positivity, symmetric and triangle inequality one by one.

Q: Let X be an inner product space over C and x, y = X are such that || x + y || = || x-y ||, then show…

A:

Q: 1, Evaluate & FdV where V S is bounded by the planes x=0, x=2₁ y = 0, y = ²,2 = 1, and F=64²³yo, 2…

A:

Q: Let f be defined and continuous on [a,b]. Then for each x E[a,b] there exists a polynomal p(x) such…

A:

Q: Let T be the co-finite topology on R and d0(x,y)=|x-y|. Prove that the function f:(R,T) to (R,d0)…

A: Given that T be the co-finite topology on ℝ and d0x,y=x-y To prove that the function f:ℝ,T→ℝ,d0…

Q: Let f be defined and continuous on [a,b]. Then for each x E [a,b] there exists a polynomal p(x) such…

A:

Q: For z ∈C, define Tz: C →C by Tz(u) = zu. Characterize those z for which Tz is normal, self-adjoint,…

A: Firstly, find out Tz*.

Q: Let f(x,y,z) = zIn(x³ycos(z)), find Z 3x² O In(x3ycos(z)) - ztan(z) 0 2 Y о 3z X af ila

A:

Q: 4. Show that ||y|| = maxxela.b]ly'(x)| defines a norm for the linear space Y - %3D {y e C' [a, b]: (…

A:

Q: 2. Assume that (X, dX) and (Y, dY ) are complete spaces, and give X × Y themetric d defined byd((x1,…

A: To prove that every Cauchy sequence in the product space XxY converges (under the metric defined in…

Q: i) True or False: T = {0, R}U {(-x, a] : a € R} is a topology on R.

A:

Q: Let (X, t) is topological space, ACX, (X,d) is a metric space. Prove that: f: (X,d) -> (IR, Ts),…

A: Let ( X, d ) be a metric space and R with the standard topology. Then, we define a function  f(x) =…

Q: Let C[0, 1] have inner product [f(x), g(x)] = ∫ f(x)g(x) dx. Find α so that (x + αx2 ) ⊥ (x + 1)

A:

Q: Let A € R"Xm, n > m. a) Use the SVD of A to deduce the SVD of AT A. b) If m = n and A is full-rank,…

A:

Q: Apply Gauss Divergence theorem for F = x²i + y²j + z²k_over the cuboid S bounded by 0 < x< 1, 0< y<…

A:

Q: et X=R^(2) and C={(x y)in R^(2):x>=0 y=0}. Show that C is not a generating cone.

A: Given: Let   and .  To show that C is not a generating cone.

Q: PROVE: If T is a linear map from V to W, then T(0) = 0.

A:

Q: Let (X, || · ||) be a Banach space, J CR be bounded and F C C1 (J, X ). If sup ||f(x)|| < , f€F, rɛJ…

A:

Q: b. Draw the image of the W-plane into which the region bounded by the straight lines in the Z-plane…

A: The solution is given below

Q: For normal linear spaces X, Y, prove that BL (X, Y) is complete if Y is complete.

A:

Q: IIC sUUset of R". tve R" and let k e R. Prove that S = {x e R":x.v = k} an affine subset of R". %3D

A:

Q: Let (X, d) be a compact metric space and T: XX be a mapping that satisfies d(Tr,Ty)<d(r,y) for all…

A: What is Metric Space: Metric space is an abstract set in mathematics, particularly topology, with a…

Q: Let X = (!,2) and T3} U = (1,2-)inel,2,U+ jult. %3D n+1 show that X is not T - space

A: This is a problem of Topology.

Q: Let X = R, and let T consist of 0, R, and all intervals of the form (a, b) where a, b e R and a < b.…

A: A topological space is an ordered pair {\displaystyle (X,\tau ),} where {\displaystyle X} is a set…

Q: Let R have its usual topology and let Y = {x, y, z}. Define f: R → Y by: %3D x 0 (х, f(x) {y, (z,…

A: Quotient topology: If X is a topological space and A is set and if p:X→A is a surjective map, then…

Q: Let R equipped with the Euclidean topology. Let Y = [1,5] and denote by Ty the induced topology on Y…

A: Definition :Let (X, T) be a topological space with topology T. If Y is a subset of X, then the…

Q: An Elliptical norm for R2 is defined by + 4u2, where x = (1,u). (i) Prove that l is a norm for R2…

A: This is a problem of Functional Analysis.

Q: Let f be defined and continuous on [a,b]. Then for each x E[a,b] there exists a polynomal p(x) such…

A:

Q: Consider E = C( [o,(]) as normed spaces with normS II f | = sup 1fcxl and l fll,=Sifwildx. %3D Show…

A: