Let X = Rand def ine a function <,> on R x R to R as < x, y >= eTY verify all conditions of inner product if it satisfies or not

Q: Show that the function a Y1 (x, y) = [21 2] в с Y2 is an inner product on R? if and only if b2 – ac…

A:

Q: Let f: R R given by f(z, 9) = (x+2y, z- y). Of is 1-1 of is not 1-1 of is not onto of is onto

A: Given that fx,y=x+2y,x-y

Q: Let ⁄2 be a bounded interval. For any g € L²(2), define the operator T, by T,(f) = f Then T, is a…

A:

Q: 3. Let SP(la, b, c,d.e) and define W:SXSS as the union function: (A, B)AUB. for ill (A. B) ESXS. (a)…

A: Given: S=P({a, b, c, d, e})Define A:S×S→Ssuch thatA(A, B)=A∪Bfor all (A, B)∈S×S We see that S is a…

Q: Let n E w. Suppose that f:n → A is onto A. Prove that A is finite.

A:

Q: Prove that the map g : Z →→ N defined by | 2k, -(2k+1), k < 0 k 2 0 f (t) = is one to-one nd mans…

A:

Q: Let f : R → R be f(r) = + 1+ \|° (a) Is ƒ everywhere defined? If not, give the domaim (b) Is ƒ onto?…

A: Given, f: R→R be                       f(x)=x1+|x|

Q: 6. Compute the norms for (a)||f()||, (b) ||F(x)||2, (c)||f(x)||, where f(x) = -V1-r", defined on [0,…

A: Given

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: Two functions u and v of x and y are said to satisfy the Cauchy- Riemann eguations if u, = v, and u,…

A: Thje given functions are,u(x,y)=y(x2+y2) , v(x,y)=x(x2+y2)

Q: then L is a linear operatorE Let L(x): R R be a mapping such that L Select one: True False

A:

Q: 2. Prove the general formulas (GF)* = G*F* and (F')* = (F*)' in the special case where F and G are…

A: An isometry in R3 is a one-one onto function which preserves distance that is for a ∈R3and b ∈R3…

Q: Let . be the operation on the cartersian product of real numbers R • R such that (a,x) * (b.y) =…

A: Given (a,x)*(b,y)=(ab,x+y)

Q: Let F(x, y,z)=xi + yj+zk, f(x,y,z)=||F(x,y,z)||. Prove that: Vf" = nf"-²F F v(In f) =

A:

Q: Find the Wronskian of the given pair of functions: X O x² et O xet i, xet

A:

Q: .

A: Let L be the linear operator on R3 defined by

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

A: Consider P2 (polynomials of degree atmost 2) with the inner product defined as follows, p, q=∫-11…

Q: Exercise 3 Define f : Rª\{0} by f(r) = |r|,'. Using the e-5-definition, prove that f is continuous.…

A: We are given a function f whose domain is ℝd-{0}. The function f is defined as  f=x2-1. Where…

Q: Prove the following: a) Let ř =. Show that V In |F| =

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: 3. Let C[0, 1] be equipped with the inner product (f. g) = f(x)g(x)dx, where f = f(x) and g = g(x).…

A:

Q: Exercise 3. Show that f : Z → N so that f (z) = z² + 1 for all z E Z is neither one-to-one %3D nor…

A: One-One Function: A function is said to be One-One if for every element in the domain there exists…

Q: A function from ZxZ to Z defined by f(m, n) = m-n is an onto function. O True O False

A: The given function is                                     f: ℤ×ℤ→ℤ  such that    f(m,n)=m-n. A…

Q: Functional theory and functional analysis

A: Consider the provided question, Let A be an invertible operator acting on space X.

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: Let x = P[a,b] be the set of all polynomials over [a, b] with subnorm 11.11.00. Using Uniform.…

A: Given that: X=Pa,b be the set of all polynomials are a,b with suspension ∥·∥∞. The objective is to…

Q: Let f(x, y, z) = x²y°z – xyz², and compute: (a)

A: In this question we will use basic of vector  calculus i.e gradient of any function and curl of…

Q: Let P denote the set of functions f in C([0,1]) with f(x) > 0 for all x. Then P is a closed subset…

A: Given that, P denotes the set of functions f in C0,1

Q: Let L be a linear operator on R1 and let a = L (1). Show that L (x) = ax for all x ∈ R1.

A: It's a highly simple question.

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

A: We have given operator A such that A: C[0,1]-C[0,1].Also we must know that the inverse operator…

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: 9. Let F : R" → R". Prove that for each a the mapping h→ dF[a; h] is linear on R".

A: Given: F:ℝn→ℝm   To prove: For each a, the mapping h↦dFa;h is linear on ℝn

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: IS V is finite dimensiona! oad f is a homomor- Ex 6. phisni of V into itself which is not onto prove…

A: Given: V is finite dimensional and f is homomorphism of V into itself such that f is not onto. To…

Q: Let A: C [0,1]→C [0,1]. Prove that operator A is continuous reversible, find A. Ax() = x() + ja-m}xt…

A: Solution:- We have given operator A such that A: C[0,1]- C[0, 1] Also we must know that the inverse…

Q: Prove A = BH (x)A = (Vx)B without using the equivalence theorem.

A:

Q: To get a feel for higher order ODES, show that the given functions are solutions and form a basis on…

A:

Q: Show that the functions x and x2 are orthogonal in P5 with inner product defined by (5), where xi =…

A:

Q: If the Wronskian of f and g is t cos t - sin t, and if u = f+ 2g, v = f - g, find the Wronskian of u…

A:

Q: Use Green's Theorem to evaluate F dr, where F(2, y) = (9xy, y" - 8) and C is the rectangle with…

A: That's easy. Have a great day!!!

Q: Let f(x) = sin x and g(x) = cos x, and W = span{f,g}. Use the Wronskian to show that f and g are…

A:

Q: Suppose that f: C C is a holomorphic function such that there is a fixed kEN for which =1, for all.n…

A:

Q: Show that if a contraction map f: from M to M has a fixed point then it's unique.

A: Definition: Let X, d be a metric space. The map T: X→X is said to be a contraction mapping if there…

Q: Prove or disprove For function f:→ R → R f(x) = x² and every onto function g(x) : [0, 00) → holds a)…

A:

Q: Suppose f : Z → Z is defined by f(m, n) = m – n and g : Z→ Z is defined by g(n) = "i. (a) Is f…

A:

Q: |A:k||2||Br:||2) < ||A||F||B||F k=1

A: answer is in next step

Q: Determine whether or not the function f : Z × (Z − {0}) → Z is onto, if f((m, n)) = ⌊m/n ⌋

A: Given that: f(m,n)=mn,  f:Zx(Z-{0})→Z

Q: Prove that if w = f(x,y,z), x=r-s, y=s-t, and z =t-r then ôw ôw ôw + âr' as = 0. ôt

A:

Q: Use th properties of linear Space A upon t to prove s a) 0.X - o for all X EX b) a.x=0 =>(a=07v…

A: