1. Let V be an inner product space. Suppose f,g E V such that f - g Lg (a) Prove that 1l12= ||f- g|2 + ||9||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: Show that the function below is not an inner product on R3, where u = (u1, u2, u3) and v = (v1, v2,…

A: We have to show that the function below is not an inner product on R3, where u = (u1, u2, u3) and v…

Q: We denote the set of polynomials with real coefficients as R[X] := {ao + a₁X + a₂X² + ... + a₂X": n…

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Q: Consider the relation R on the set of functions defined by (f, g) = R if and only if f(x) is…

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Q: A relation < is defined on R? by (x1, x2) < (y1, y2) if and only if x1 < yi and x1 + x2 2 yı + y2.…

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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: Consider the function T:R² → C defined by T(x,y)=x+iy. Show that this is an isomorphism of real…

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Q: 1. Let P3 be the space of polynomials of degree at most 3, i.e., P3 = {po+ p1x + p2x² + p3x% : po,…

A: Given a vector space, P3=p0+p1x+p2x2+p3x3 |p0, p1, p2, p3∈ℝ  To prove that A=1,1+x, 1+x2, 1+x3 is a…

Q: Q;:Is the following mapping =|A + B| an inner product on M2(R).

A: No it is not a inner product.

Q: Let f: Z/4Z → Z/2Zx Z/2Z be defined by f([0])= ([0], [0]), ƒ([1]) = ([1],[1]), f([2]) = ([0], [1]),…

A: We see,         [2] = [1] + [1]    but, f([2]) = ([0], [1])          f([1]) + f([1]) = ([1], [1]) +…

Q: let N be the set of natural numbers and f:NU {0} →NU [0] be definedby: (n+1 when n is even f(n) =…

A: Prove that the function f is injective for x∈N. \ The function is injective as the function is…

Q: If {v, w} is independent, find conditions on the scalars k1 and k2 so that the set {k,v + w, v +…

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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: (5) Let ƒ : A → B and g : B → C, where f and g are functions and A, B, and C are sets. (a) Prove…

A: In 2 nd part construct a function

Q: Let f: R → [1, ), f(x) = x² + 1 g : [1, 0) → [0, ∞), g(x) = Vx – 1 i) Derive g • f and show your…

A: Composition of two functions

Q: Justify wether the following is an inner product or not =5x1y1-x1y2-x2y1+2x2y2 where x=(x1,x1) and…

A: By using axioms of inner product we have to show whether &lt;x, y&gt; is an inner product or not.

Q: (b) Let U, T be usual and cofinite topologies on se:t of real numbers R and f be a map from R into R…

A: As per our guidelines we are supposed to answer only first question. So i have solve  question (b)…

Q: Find all invertible meromorphic functions defined on the extended complex plane C* that map x-axis…

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Q: g) for any bounded

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

A: This is a problem from topology.

Q: 2) Let X be normed space and xo + 0 be an element of X. Then using Hahn-Banach Theorem (Normed…

A: We will construct a functional satisfying given data

Q: Show how a real

A: To Prove that how a real linear functional u on a complex linear normed space gives rise to a…

Q: Prove that the mapping f : U(16) ---> U(16) given by f(x) = x^5 is an automorphism.

A: The given map is f:U16→U16 defined by fx=x5. We have to prove that f is an automorphism. We know…

Q: Let A: X₁ ≤X → Y be a closed operator where X and Y are Banach spaces. Define ||x|| = ||x||+||Ax||,…

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Q: Consider the function g : ZxZ → Z × Z defined by g(x, y) = (4xy, 9x + 5y). (a) Is g injective? (b)…

A: (a) we see (5, 0) and (0, 9) are different elements in ZxZ.      But, g(5, 0) = (4.5.0, 9.5+5.0) =…

Q: (a) Define ƒ : R –→ Z as follows: f(r) is the smallest integer y which is at least r. Decide whether…

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Q: Which functions are one-to-one? Which functions are onto? Describe the inverse function for any…

A: One to one: A function f : A → B is one-to-one if for each b ∈ B there is at most one a ∈ A with…

Q: Let zo + 1 be a zero of order 3 of an analytic function g(z) and f (z) = (z-1)2 (g(z))³" Then z, is…

A: Use the concept that if f(z) has zero at z0 of order m then 1/(f(z)) has pole at z0 of order m. By…

Q: 1.7) Example: Let P (R) the vector space of real functions , define by: = f(x)g(x) dx Is inner…

A: Given: Pn(ℝ) the vector space of real functions, define f, g by: f, g=∫01f(x)g(x)dx Is inner…

Q: 4) Let ƒz = dƒ/əz and ƒz = 0ƒ/əz. Recall ƒ = u + iv, the notation f is used for the complex…

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Q: 4) Consider the function g(x) = [2r – 1] where g : R → Z. a) Is g a one-to-one function? Explain. b)…

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Q: (b) Let the function g : Z → Z be defined by g(x) = 7x + 3. (i) Show that g is one-to-one. (ii) Show…

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Q: Suppose that the function f : D → R and g : D → R are uniformly continuous and bounded. Prove that…

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