a. Let X and Y lee nonempty nets and f:X→Y. ASX and B S Y, thew fLA] nB = f [An f"[B].

Q: Let w = xydx + 2xdy - ydz E 2'(R) and let a: R a(u, v) = (uv, u², 3u + v). Compute do, a*o, d(a*o),…

A: Given ω=xy dx+ 2x dy - y dz in Ω1ℝ3x,y,z And let α:ℝu,v2→ℝx,y,z3 be given by  αu,v=uv, u2, 3u+v. To…

Q: Show that the subset of R2 given by {(x, y) E R² : xy 1, x > 0} is closed.

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: If there exists open ball B (a, r) such that B (a, r) NG= Ø then a is not limit point of G True O…

A: TRUE

Q: Let (X, F) be a o – algebra | , and be two probability measures on (X, F) so that both are ergodic…

A:

Q: (a) Show (by calculation) that {u1(x), u2(x)} is an orthonormal set. (b) Find the projection p(x) of…

A: Given the space C0,1 of continuous function on the interval 0≤x≤1, with the inner product defined as…

Q: The main point of this exercise is to use Green’s Theorem to deduce a special case of the change of…

A: We will use the basic knowledge of multivariate functions to answer this question. Note that in your…

Q: 5. Prove that the baker map 2x B(z) = 2z - 1 <z<1 is chaotic on (0,1).

A: Given the baker map, Bx=2x,0≤x≤122x-1,12≤x≤1

Q: Let XC Rn convex. Prove that x is an extreme point of X if and only if the set Cis convex, where X =…

A: Let X⊂ℝn be convex . To prove: x is an extreme point of X if and only if the set C is convex, where…

Q: let H = {n² |n E z*}, determine whether it is closed under

A: Given : H={n2|n∈Z+}   To Check:  H is closed under binary addition or not

Q: Suppose {u1, Span{u;}nSpan{w;}={0}. Ur, W1, Ws} is a linearly independent subset of V. Show that •..…

A:

Q: let H = {n² |n e Z*}, determine whether it is closed under

A: Here, we have H=n2| n∈ℤ+. To show the set is closed under binary multiplication. This means we have…

Q: Let M1 = ( S1, Ih) and M2 = ( S2, I2) be two matroids with disjoint ground %3D sets. Prove that the…

A: Given: Two matroids M1=S1,I1 and M2=S2,I2 with disjoint ground sets.   To prove: The subset system…

Q: Show that the subset of R2 given by {(x, y) E R² : xy = 1, x > 0} is closed.

A:

Q: * Let Z be with the discrete topology.11 If A = {1,3,5, 7,...}, then Aº %3D %3D A O N O R O

A:

Q: Find the fowien senies fon | X), -T<X <A

A:

Q: Exercise 8.2. Let X, Y, Z be normed spaces and suppose that T E B(X, Y) and S e B(Y, Z). Show that…

A:

Q: Show that a set {V1, ..., Vp} in R" is affinely dependent when p > n + 2.

A:

Q: Evaluat..f.x.dA. wha.is.the..sgiem.in.the.fiigt. R quandrant..bounded..xy=16. X..8....y.z.. y.e.x...

A:

Q: Let v: DX be a net in X. Then a point xo X is a cluster point of a net v = {xa} if and only if there…

A:

Q: (b) If X is a normed line- ball M= {xeX:x≤1}

A:

Q: 6. Consider S = [a, b] with p(x, y) = |x – y|. Verify that T : S – S with T(x) = - and a > 1 is a…

A:

Q: The orthogonal projection of y onto u is the same as the orthogonal projection of y onto cu whenever…

A: solution

Q: Let A: C [0,1]→C [0,1]. Prove that operator continuous reversible, find A. 2.11. Ax(t)= x(1)- -…

A: The given operator A such that A:C0,1→C0,1. We use the inverse operator method for solving linear…

Q: Given any Hermitian space E with › E, if (f(x), x) = 0 for all x € E, th I

A:

Q: Let {Aq | a E I} and {B3 | BEJ} be two indexed collections with nonempty index sets I and J. Prove…

A:

Q: Extend the linearly independent Set S={x+1, 2X²} to the space base P₂ (R)

A: We are given a linearly independent set S= {x+1, 2x2} We have to extend S to basis of P2(R).    Let…

Q: Let v e R" and letk e R. Prove that S = {x € R" :x•v = k} is an affine subset of R". %3D

A:

Q: Let X be a normed space and A € BL(X) be of finite rank. Then σe (A) = σ₁(A) = o(A).

A: 1

Q: (b) Consider X = C[0, 1] with the norm %3D = max |x(1)), re[0,1] and let T: X X be the mapping given…

A:

Q: Compute the norms for (a)||f(1)||, (b) ||F(x)|l2 (c)||f(x)||, where f(r) = -VI-r, defined on [0, 1).

A: Given- fx =-1-x defined on [0, 1] (a) we have to calculate fx  fx=∫01fxdx        =∫01-1-xdx…

Q: 2. Let S,T C R" with S being a proper subset of T. If span(S) = span(T), then T is linearly…

A:

Q: Let S be a nonempty bounded subset ofR. If for every e > 0, there exists an x E S such that x > b -…

A:

Q: 4. Let f : R" → R", x+ Ax + b be an affine function, where A E R"xn,beR". (a) Show that if X CR" is…

A: Let f: ℝn→ℝm be an affined map defined as x↦Ax+b, where A∈ℝm×n and b∈ℝm. Then it follows: fx=Ax+b…

Q: . Show that if S is a subbasis for topologies T1 and T2 on a space X, then T1 = T2.

A:

Q: Consider Uu=4uzzy u(x,0) = u(x), u₁(x,0) = u₁(x), on the domain -∞<x<∞, 0≤t<∞o. Suppose that both…

A:

Q: of a space X and let x e X. Then x e A iff there exists a net in A (that is a net whi to x in X.

A:

Q: Let, T: R' → R';T(x, y,z) = (x+ y+ z,2y + z,2y +3z) Find the dimension of KerT and ImT

A:

Q: Consider R? and R3 with the Euclidean (c) norms. Compute || T || for T : R2 →R3 defined by T (x1,…

A:

Q: 2 Let A be a fixed nonempty subset of set X and I= {Bc X | A Z B}. Show that (X,I) %3D is a matroid.

A:

Q: prove that Sa/Gn has f discription with M-ly ny-l Degrees tof freedom

A: Suppose that xi and yi are two random samples of size n, drawn from a normal population.  Population…

Q: Which of the following nets will form a cylinder? А. В. С. or D

A: Follow the procedure given below.

Q: Iw (X, D) discrete space A = {2,3,4s A A Bd(A), Euct CA)? EXercise %3D Fined

A: Given that A=2,3,4 The aim is to find A,A°, Bd(A), Eucd(A) A=2,3,4 And, A°=∅ Here A is closure of A…

Q: (3) Suppose that T: V F is a linear map. Prove that if u E V such that u 4 Ker(T) then V = Ker(T)…

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 = (!,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: Prove that the map || - || : L(V, W)→R defined by ||L|| = sup {|| L(x)|| | ||| < 1} is a norm on…

A:

Q: (a) Prove that x, y = -1, y = 1 1 (b) Prove tha and n: m and

A: This is a problem of Number Theory.

Q: If e-ispure imaginary W hat Restriction is placed on z?

A: