Let X# and t1, t2 are two topologies on X such that 11CT2. Prove or disprove that if (X,t2) is a T2-space then (X,t1) is a T2-space.

Q: Let T be a linear operator on an inner product space V. If (x), y>= 0 for all x, y ∈V, prove that…

A: Consider a basis for V as follows.

Q: 1. Provide an example of a vector space Mmxn(Q) that is isomorphic to the vector space P19(Q).

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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: Suppose that the set A ={X₁, X2..., Xn}, X₁ ER" spans the entire space R", i.e. for any arbitrary n…

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Q: Let P3 be the vector space of polynomials of degree at most 3, and let S = {p1(x), p2(x), p3(x), p…

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Q: Consider the real space R3 provided with the usual inner product. Given the elements u=(1,1,1) and…

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Q: Suppose V₁, Vm, and W are vector spaces. Prove that L(V₁ × ... X Vm, W) are isomorphic to L(V₁, W) ×…

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Q: Let V be an inner product space, and suppose that T: V→V is linear and that ||T(x)|| = ||x|| for all…

A: Given:

Q: Q3 Prove : If V is a normed linear space, and WcV, and U a closed subset of W Then given ve V,3u"…

A: given if V is a normed linear space, W⊂V and U is a closed subset of W then for v∈V there exist u*∈V…

Q: 10. Let (X, 71) be a Lindelöf space and (Y, T2) be a compact space. Then show that the carte- sian…

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Q: Let V, W be finite-dimensional inner product spaces and T € L(V, W). Prove: (a) If T is injective,…

A: Injective Function:  A function is said to be injective if each element of the codomain is mapped to…

Q: Let T be a linear operator on an inner product space V, and supposethat IIT(x)ll = llxll for all x .…

A: Let us consider T be a linear operator on an inner product space V, and suppose that IIT(x)ll =…

Q: Q4) Let C(R) be a normed space such that || f -sup{ Ifl,x E R}and T: C (R) -R define Tf)-If I. Is T…

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Q: Consider the vector space P, of all polynomials of degree = a, a,+b, b2 , then the inner-product of…

A: Given polynomials are: p(x)=a1+b1xq(x)=a2+b2(x) Inner product function is defined as:…

Q: Prove that if X is a Banach space, X ∕= {0}, then there is a non-zero biounded linear functional on…

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Q: Let (X₁, || ||1), (X2, ||· ||2) be Banach spaces. On X₁ X X₂ define || (X1, X₂) || = || X1 ||1 + ||…

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Q: Let X & Y be normed spaces & x # 204. Then BL (X₂Y) is a Banach space in the operator norm if and…

A: Here BL(X,Y) (or, B(X,Y)), is the space of bounded linear operators from X to Y. First ,we show that…

Q: consider the relative topologies of the usual topology on R on I₁ = (2,4) and I₂ = (8,16) prove that…

A: Here f is a function from I1 to I2 defined as f(x)= 4x for all x in I1. Where I1=(2,4), I2=(8,16).…

Q: Which of the following is true about the subset of polynomials of degree 2 or less?…

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Q: if T is an operator in a real vectorial space with characteristic polynomial (x²-2x+5)². What are…

A: For the solution follow the next steps.

Q: Find a basis {p (x), q (x)} for the vector space {f(x) = P3[x] | ƒ' (−1) = ƒ (1)} where P3[x] is the…

A: We have to find p(x) and q(x).

Q: If scalar multiplication and addition is redefined on R² in the following way, show that it is NOT a…

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Q: Consider the vector space P, of all polynomials of degree = a, az+b, b2, then the inner-product of…

A: If p(x)=a1+b1x and q(x)=a2+b2xThen<p,q>=a1a2+b1b2

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: Theorem 4.7. (1) A T2-space (Hausdorff) is a T1-space. (2) A T3-space (regular and T¡) is a…

A: 1)  Consider X is a T2 - space. To prove that, X is a T1- space. By the definition of T2- space,…

Q: Find a norm for C([0, 1]), so that C([0, 1]) is not complete under this norm.

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Q: 12. Suppose that a, b e R and a < b. Show that the vector space C[a, b] of all continuous complex…

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Q: Prove that the Joukowski map 1 p(2) : (z + maps ONTO T - (-1, 1)

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Q: Suppose V is a 4-dimensional vector space, and that T is a linear operator on V with minimal…

A: Dimension of Ker T

Q: Let (X, d) be a compact metric space and let A ⊆ C(X; C) = {f : X → C; f is continuous} be an…

A: please see the next step for solution

Q: Consider the vector space P, of all polynomials of degree = a, a2+b, b2 , then the inner-product of…

A: We have to find the inner product of the following -  p = 2 + 1.7x q = 4.8 + 3.2x

Q: 3. Prove that in any normed linear space, xo is the Chebyshev center of the closed ball {x : ||x –…

A: Given: Given any normed linear space x

Q: Let P2 be the vector space of all polynomials of degree ≤ 2 with coefficients in R, and S= {1 + 2x,…

A: The basis for the vector space P2 is 1, x, x2. We have to write the matrix form for the set S. We…

Q: Either use an appropriate theorem to show that the given set, W, is a vector space, or find a…

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Q: im(T1 + 14T) im(14T† + T2).

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Q: Either use an appropriate theorem to show that the given set, W, is a vector space, or find a…

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Q: Show that the set of all pairs of real numbers (x, y) with the operations (X1, Y1) + (x2, Y2) = (x1…

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Q: Let V be an inner product space, and let y, z ∈V. Define T: V →V by T(x) = &lt;x, y&gt;z for all x…

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Q: 4) Suppose that T is an operator on an inner-product space V such that T2 = T. Prove that T is an…

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Q: 2. If T1, T2 are normal operators on an laner produce space with the property that either commutes…

A: Given that T1, T2 are normal operators of an Inner product space. So, T1T1*=T1*T1 and T2T2*=T2*T2,…

Q: If p4 is a vector space formed by polynomials of degree less than 3 And  s = { ax^3 + bx^2 +cx + d…

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Q: Assume that T : C° → C is a linear operator with minimal polynomial m7 (x) = (x – 2)2 . Find all…

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Q: , Find a basis {p(x), q(x)} for the vector space { f(x) e P3[x] | f'(-5) = f(1)} where P3[x] is the…

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Q: Suppose that the set A={X₁, X2..., Xn}, X₁ ER" spans the entire space R", i.e. for any arbitrary…

A: Given: The set A=x1,x2,...,xn, xi∈ℝn spans the entire space ℝn, i.e. for any arbitrary n-vector…

Q: Define A : L3/2[–1, 1] → C by Prove that A is a bounded linear map and compute its norm.

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Q: 7. Let V be a vector space and l: V → R be a linear map. If z E V is not in the nullspace of l, show…

A: Given: 7) V is a vector space and l:V→ℝ is a linear map. To show: If z∈V is not in the null space of…

Q: Every T2 (Hausdor ff) space is T1 space. True O False O

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Q: Let T be a linear operator on an inner product space V, and suppose that ||T(x)||= ||x|| for all x.…

A: According to property of norm the value of ||x||=0 iff x=0.

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.