Consider L[-1, 1] with the essential sup-norm (see Definition Question 1. 3.2.12). Show that the essential sup-norm cannot be induced by an inner product.

Q: Prove that subring R[x2, x3] not isomorphic to R[x].

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Q: Show that a collection {(-∞,q) C R | q is rational } is a basis for a topology on R.

A: Since for your B part there is no information regarding the  topology homework 2 problem 4. Hence if…

Q: Verify that the Galois imaginary is primitive in GF(3, x³ + 2x+1)

A: Given:- Verify that the Galois imaginary is primitive in GF(3,x3+2x+1)

Q: Show that for any two finite-dimensional L-representations V and W of L: (a) Hom (V, k) ~V* as…

A: To show that for any tow finite-dimensional L-representation V and W of L HomkV, W≃V*⊗W as L-module

Q: 10. Prove that T, is topologically conjugate to the quadratic map F(z) = 4z(1 - z).

A: Solution of 10. To prove that T2 is topologically conjugate to the quadratic map F4, we have to find…

Q: Prove that the effective factor is a Hermitian factor in a specif and x-i

A: Momentum operator is defined as an Hermitian generation of translations, then it is hermitian by…

Q: prove this: The minimal sigma field contain C is unique

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Q: Proposition 6. Consider the function g : Z² → Z² defined by g(m, n) = (m - function g is a…

A: Consider the function , g : ℤ2 → ℤ2 defined by ,  g(m,n) = (m - n , n - 2m). We need to show that ,…

Q: Which of the following form an orthonormal set in a euclidean inner product space R3 ? 1 В. -- 3 3…

A: Orthonormal set of vectors:    The vectors whose norm is 1 and whose dot product is 0 is known as…

Q: c) The operation (ao + a1r + azx², bo + bịx + b,x²) = aobo + a¡b1 – azbz defined on P2 has all the…

A: We have to check whether the operation defined on P2 satisfies all the properties to be an inner…

Q: Let T and U be self-adjoint operators on an inner product space V.Prove that TU is self-adjoint if…

A: We are given that T and U are self-adjoint operators on an inner product space V. We need to Prove…

Q: Show that for any two finite-dimensional L-representations V and W of L: (a) Hom (V, k) ~ V* as…

A: As per bartleby guidelines for more than one question asked only one is to be answered please upload…

Q: 4. Show that the collection C = {[a, b) | a < b, a and b rational} is a basis that generates a…

A: Given:         l = {[a,b]} | a&lt;b , a and b rational}

Q: In an inner product space, if , is not orthogonal to , and v is not orthogonal to w, then ū is not…

A:

Q: 4. Exercise §5.2 #20 [This exercise presen Theorem 5.9 using the Möbius Inversion Form to that in…

A: Given, d be a positive divisor of p-1. To prove that                            f(d)=ϕ(d)

Q: Prove that the following does not define an inner product R?: ((a, b), (c, d)) = ac – bd. on

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Q: Show that the maximum norm on C[a, b] is not induced by an inner product

A: to show that the maximum norm on Ca,b is not induced by an inner product . as we know , a norm X,·…

Q: Suppose X and Y are normed spaces and X + 0. Prove that BL(X, Y) is a Banach space with the operator…

A: Given, X and Y are normed spaces and X ≠ 0. We have to prove that BLX , Y is a Banach space with the…

Q: Prove that if A is invertible and BA = CA, then B = C.Give a counterexample to show that the result…

A: Given: A is invertible and BA = CA  To prove: B = C

Q: 2. Prove that if 5BA+ ACA – I = A, then A must be invertible.

A:

Q: Prove or disprove that (R, Tco-finite) is T2- space

A: It is given that ℝ,τco-finite. τco-finite=U⊂ℝ   such that  X-U is either finite or X  To check…

Q: 32. Prove statement d of Theorem 3.9: If G is abelian, (xy)" = x"y" for all integers n. bns liD to…

A: I have proved using mathematical induction

Q: Prove that if A is invertible and AB = O, then B = O.

A: To prove that if A is invertible and AB=0, then B=0.

Q: a) Show that this definition of the inner product satisfies both of the above conditions.

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Q: Define g : (-1, 1) → R by g(x) 1- x2 " Show that |(-1, 1)| = |R| by showing that g is a bijection

A: Concept Used: For a function f:A→B If f is a bijection, then |A|=|B| Consider the function g:(-1,…

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

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Q: If A is 3 × 3 with rank A = 2, show that the dimension of the null space of A is 1.

A: Given: A is 3*3 matrix with rank A = 2

Q: Prove that det(At) = det(A) for any A ϵ M2x2(F).

A: Suppose we are given a square matrix A with four-element: a,b,c,d A=abcdAt=acbd

Q: 32. Show that, in a normed space, x x implies -(x1 ++Xn) → x.

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

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Q: Prove that K4 is not outerplanar

A: In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices…

Q: 1. Show that the definition (x1, x2) · (Y1, Y2) = x1Y2 is not an inner product on R?.

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Q: Define which of the followings are ring homomorphisms from M2(Z) to Z. (:)- a (a) Ha (projection…

A: Ring homorphism

Q: 1. Prove that any quasigroup satisfying one of the identities {(ry)x satisfy the other. = y, z(yr)…

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Q: Let A find on Conditions a,bic and d has infinitely that AX =o many solution . what must there Pere…

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Q: Prove that T is a lincar transformation.

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Q: The union of two ideals is ideal* The only idempotent in an integral domain is 0 O T O F O O

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Q: Prove that Z[x] is not a principal ideal domain.

A:

Q: 8. Show that A is Hermitian if and only if A = B - C with B,C positive.

A: A is Hermitian if and only if A=B-C with B, C positive.

Q: 10. If A is 3 x 3 with rank A 2, show that the dimension of the null space of A is 1.

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Q: Let ƒ : [0, 1] → R be given by f(1) = xr and g : R → [0, 1] be given by g(x) = Prove that |[0, 1]| =…

A: Cantor-Schrӧder- Bernstein Theorem: if there exist injective functions f: A → B and g: B → A between…

Q: Let f : [0,1] → R be given by f(x) = x and g : R → [0, 1] be given by g(x) = . Prove that |[0, 1]| =…

A:

Q: Suppose p > 0. Prove that there is an inner product on R2 such that the associated norm is given by…

A: For first,Let's assume that p=2, then there exists a inner product on IR2 that is given by euclidean…

Q: Let ƒ : [0,1] → R be given by f(æ) = x and g : R → [0, 1] be given by g(x) = Prove that |[0, 1]| =…

A: We know that, Cantor-Schrӧder- Bernstein Theorem: if there exist injective functions f: A → B and g:…

Q: Prove that if A is invertible and AB=0, then B=0.

A: Given:

Q: Let ƒ : [0, 1] → R be given by f(x) = x and g : R → [0, 1] be given by g(x) = Prove that |[0, 1]| =…

A: Cantor-Schrӧder- Bernstein Theorem: if there exist injective functions f: A → B and g: B → A between…

Q: (1) is a Hilbert space when the inner product is defined by ((an), (δη)) = Σanon. nEI

A:

Q: 10. If A is 3 x 3 with rank A = 2, show that the dimension of the null space of A is 1.

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Q: Exercise 3.2.3 Show that Zg under addition is isomorphic to Z; under multiplication.

A: We know that Z6 = {0, 1, 2, 3, 4, 5} and Z7* = {1, 2, 3, 4, 5, 6}. A group isomorphism is a function…

Q: 6. Find an explicit function f: [0,1] and prove why it is (0, 1) that is a → bijechove bijection

A: