consider the space generated with the set of functions  {x, x^2} , show that the set of elements of the base are orthogonal in the interval. [−1,1]

Q: Define an inner product space for continuous functions on [0, 1] as (S, g) = [f(x)g(x)ax with the…

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Q: Use Hahn-Banach Theorem to show that a normed linear space X is finite dimensional if its dual X' is…

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Q: В. B. Show that the function f (z) = -i z2 is analytic.

A: The solution is given as

Q: 5. Determine whether the function f : R+ → R defined by the rule f (x) = positive real numbers x in…

A: A function is said to be bijective if the function is both one one and onto. The given function is…

Q: Show that the function f defined by (r, y) = (0,0) f(x, y) = { 22 + y (r, y) # (0,0) x + y

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Q: Find the derivati ve of the function at the given point in the direction of A. f(x, y) %3DIn(6x -…

A: We need to find unit vector in the direction of A vector and gradient vector at point (10,-10)

Q: Explain why the Lp[0, 1] spaces are complete normed linear spaces. What is the norn of function f(x)…

A: This is a problem of functional analysis.

Q: Use the cauchy-riemann conditions to determine whether the function f (z) = ((x ^ 3) + x(y ^ 2)) + i…

A: Cauchy-Riemann  : Let f(z)=u(x, y)+iv(x, y) be a complex valued function. Then the Cauchy-Riemann…

Q: → [0, ∞) defined by f (x) = Vx + 2. Prove that f is 17. Consider the function f: (-2, 0) surjective.

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Q: Use the Squeeze Theorem to show that f(x) = x² is in R[0, 3]. %3D

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Q: Discuss the continuity of f(x, y, z) = 1 / (x2 + y2 − z) .

A: Here The function is discontinuous when the denominator is zero. Thus, That implies

Q: Prove that a decreasing function f on [a,b]is integrable.

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Q: 12. Show that the set of all continuous function on the interval [-1,1] is a vector space under…

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Q: Let F = <5x^2, 2(x+y)^2, 9(x+y+z)^2> Find divF (divergence)

A: First of all, consider the vector F and find F1, F2, and F3.

Q: Find the Wronskian for the set of functions.{1, x, ex}

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Q: Determine the are bounded by: y2-8y+8x=0, and x=0

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Q: (v) Prove that if the space (X,T) has the fixed-point property and (Y, T1) is a space homeomorphic…

A: Given: v) To prove the given statement as follows,

Q: With X a topological space, is the function defined as A:X →X × X,x → (x, x) continuous? Investigate

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Q: Let f: X_nte, Y be X,Y are Tj spaces. show that: Xö locally compact iff Y is so . perfect function,…

A: Given: f:X→ontoY is a perfect function, where X,Y are T2-spaces. To show that X is locally compact…

Q: The Sandwich Theorem for functions of two variables states that if 8 (x, y)s f(x, y)sh(x,y) for all…

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Q: Let f(x) = Vx. Compute the difference quotient for f at a = 7 with h = 4. %3D %3D

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Q: Find and sketch the domain of f(r, y) In(r - y) 6. -

A: Given:      f(x,y)=lnx-y9-x2-y2·x2+y2-4    Interval notation : -3,0∪0,3   Set builder notation:…

Q: Let f(x) =esin 3x + 2x a. find the linearization of f(x)=esin3x+2x at x=0. b. Use your linearizato…

A: a). Given f(x) =esin 3x + 2x f(x) ≈ L(x) = f(a)+f′(a)(x−a). where the function L(x) is called the…

Q: Find a bijective function f : R^+ → R^+ different from the identity function and has the property f…

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

Q: Is the scalar valued function f(x)=(x^2-1)/(x+1) continuous?

A: This is a problem related to limit and continuity of the function. Please follow the procedure given…

Q: Q2. Show that F(x) = 3 – x², x € is contractive on its domain, but does not have any fixed points.…

A: To show - Show that Fx = 3 - x2, x ∈ -14, 14 is contractive on its domain, but does not have any…

Q: Discuss the continuity of f(x) = {Vx, x > 1 1, x<1 at x = 1.

A: A function f is said to be continuous at some point a if the value of the function at that point is…

Q: 5. Determine whether the closed interval [a,b]is homeomorphism to the closed unit…

A: This is a problem from topology.

Q: Q3/ 1- Let f: Z, → Z10 define by f(x) = 2x+2. Is fhomomorphism or not, explain your %3D answer.

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Q: Show that f : ℝ → ℝ, f(x) = 5x − 3 is a bijective function.

A: The given function is f:ℝ→ℝ defined by fx=5x-3. We say that a function f is bijective if it is both…

Q: Let f(r. y) = * if (1, y) # (0,0) if (1, y) = (0,0) Show that af/ðr and af /ðy exists at (0,0) but f…

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Q: Without the aid of Wrongskian, determine whether the given set of functions is linearly independent…

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Q: b.) answer. Let f = x² and g = (1-x)². Are x and y orthogonal in this inner product space? Justify…

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Q: For all functions y = f(x) defined on [0,1], the set f([0,1]) is a bounded set. Is this statement…

A: For all functions y = f(x) defined on [0,1], the set f([0,1]) is a bounded set  Given statement is…

Q: Determine whether the functions is a bijection from R to R. f (x) = (x + 1)/(x + 2) +

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Q: Verify that Rolle's Theorem can be applied to the function f (z) = 23 – 10z2 +27x – 18 on the…

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Q: Find all values of a and b that make -a - x, x< -2 /(x) = { Vx + b, bx² + 1, -2 Sx<1 x21 continuous…

A: Given:             fx=-a-x,            x&lt;-2bx2+1,    -2≤x&lt;1x+b,        x≥1

Q: Find and describe the points in the space that the function f(x, y,z)=In(xyz) is continuous.

A: We have to find and describe the points in the space that the function f(x,y,z)=ln(xyz) is…

Q: Determine whether the function flz) = e² is analytic or not using the Cauchy- Riemann theorem.

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Q: Determine the are bounded by: y2-8y+8x=0; and x=0

A: The polynomial y2-8y+8x=0 can be rewritten as x=8y-y28. The function is graphed to understand the…

Q: Let f (x, y) = e-(x²+y²), Find the maximum and minimum values of f on the rectangle given below.…

A: We have to find the maximum and minimum value of the given two variables function.

Q: The conjugate space H* is also a Hilbert with space espect to the inner product defined by (fx. fy)…

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Q: Suppose that f(x) is convex on an interval I. Show that f(x1) + f(x2) + ·+ f(xn) n for arbitrary n…

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Q: For the set of functions {sin(x), sin(2x), sin(3x), ...} = {sin(nx)}, n = 1,2,3, ... on the interval…

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Q: Let f : [0, ∞) → [0, ∞) be the function defined by f(x) = √x. Show that f is a bijection.

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Q: Discuss the continuity in R3 of the function f(x, y, z) %3D x2+ y2+z2 -1

A: Given: f(x,y,z)=1x2+y2+z2-1.

Q: Prove that f(x) = x 3 - l2x is differentiable. Compute f'(x) and find f '(-3), f '(0), f' (2), and…

A: Given information: The function is f(x)=x3-x. Calculation: We need to find the value of f'(x) and…

Q: Q#1: Discuss Application of inner product spaces with examples.

A: Important Application of Inner product spaces is to find Least squares approximations for a…