Prove that L∞[a, b] is complete with respect to the norm || · ||∞.

Q: 1. Suppose that x is a non-zero vector in a normed vector space V over F, and that a > 0. Find a…

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Q: 1. Prove that for every non-gero normed space X there is a non-zero bounded linear fmetional acting…

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

A: Given G and F re isometries in ℝ3 To prove  1.GF*=G*F*2. F-1*=F*-1 Since F and G are isometries in…

Q: Prove C(x) with the supremum norm is a metric space by showing the supremum norm is a metric

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

A: We will prove the given statement.

Q: Prove that if the space X' of a normed linear space X is separable (as a metric space) then X itself…

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Q: Let V be an inner product space, and suppose that x and y are orthogonal elements of V. Prove that…

A: Given:

Q: Qu:- Prove that funcdiod U = ay - 2x -2 xy-j+x* is hav monic . Firel - funchion v such thut fcas…

A: Solution

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: Which of the following form an orthonormal set in a euclidean inner product space R 3 ? 1 2 D. 2 1 1…

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Q: Let (X1, d1) and (X2, d2) be separable metric spaces. Prove that product X1 × X2 with metric…

A: A matric space X is separable if it has a countable dense subset.

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

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Q: Consider E =IR as a vector space. Show that: A function f: E → IR is a norm if and only if fu) =klxı…

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Q: 1. Prove that V is a vector space. V = pn , the set of polynomials with real coefficients and any…

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Q: 4. Consider the real vector space R?. For every x = (11,12) E R?, let ||x|| = |11|+ |r2l. a) Show…

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Q: 3. Prove carefully that the subset of R {(x1, x2, X3, ...) | x; is a nonzero real number for only…

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Q: Prove that Let T be a linear operator on V, and suppose that the minimal polynomial for T is…

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Q: Let X be a normed space. (a) If E₁ is open in X and E₂ C X, then E₁ + E₂ is open in X. Proof: (a)…

A: It is given that X be a normed space, E1 is open in X and E2 be any subset of X. We have to explain…

Q: Let X be a normed space, Xn, Yn EX such that Xnx, Yny, then 1-Xn + Yn → 4- ||xn-ynll→→llx - yll.…

A: Given that X is a normed space, xn, yn ∈X such that xn→x, yn→y. To find1. xn+ yn→x+y 2. λxn→λx ∀λ∈F…

Q: Let (X, |-||) be a normed space. Prove that if X has a Schauder basis, then X is separable. (Here,…

A: Schauder basis: If a normed space X,. contains a sequence en with the property that every x∈X, there…

Q: Let β be a basis for a finite-dimensional inner product space. (a) Prove that if <x, z>= 0 for…

A: Let β be a basis for a finite-dimensional inner product space.

Q: 1. Let k -k x = VE Find the Euclidean norm of x. Don't assume any specific value for k. So, the…

A: Since not a particular question asked, as per guidelines solution to only first question(Q1) is…

Q: Prove by contradiction the scquence {(-1)"+} is divergent.

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Q: Prove that if A is invertible, then AT is invertible. What is (AT)-1?

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Q: (a) Prove that the metric space (2,11) is complete,

A: Complete Metric Spaces:- A Metric Space (X,d) is said to be complete iff every Cauchy sequence in X…

Q: Show that Z[V-2] is a Euclidean Domain with respect to the norm N defined by N(a + by-2) = a² + 2b².…

A: This is a problem of abstract algebra.

Q: ) Let X be a finite dimensional norm space. Then prove that M = {x EX |||x || <1} is %3D compact.

A: Sufficient to prove that M is closed and bounded

Q: Computed( u, v) relative to the Euclidean norm, the sum norm, and the max norm.

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Q: Theorem: Let A and B be R-dibmodules af left R-modules M and Nrespectively. Then, MXN AXB

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Q: 3. Given the functions f, (x) = x and f2(x) = sin2x on the interval - %3D (a) Find the norm of f, on…

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Q: 0:1 Show that norm ||x|| = (Elx¡l?)ž satisfies the conditions of normed space. %3D

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Q: t & · &1 and & · &2 be two equivalent norms on a vector space X. Prove that a sequence % xn & in X…

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Q: (13) Show that | l is a norm on the space L [a,b].

A: Please check step 2 for the solution.

Q: Prove that || · ||#'(N) is a norm in the space H'(N) (u, v) H¿ (1) = /o Vu · Vv ||u||HA(2) =…

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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: Let X and Y be normed linear spaces and F:X-Y. If for every Cauchy sequence {x, in X, the sequence…

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Q: Prove that the vector space C of all continuous functions from R to R is infinite-dimensional. Shew…

A: We have to prove given statement:

Q: Prove or disprove: There is an inner product on R? such that the associated norm is given by ||…

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Q: (ii) Prove that W is a vector space of R°.

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Q: prove that A is invertible and A = B = C. %3D

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Q: Consider the linear space C[0, 1] equipped with the norm ||f || = max |f(t)|. te[0,1] Prove that…

A: Given: The linear space C0,1 equipped with the norm, f=maxt∈0,1ft   To prove: There is no inner…

Q: Q3) Let L = Z* and |x ||= max {]x1 - x3l, x2l, 15x,1} Vx = (x1, X2, X3, X4) E Z*. Is (Z*, ) is a…

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Q: 2. Let P denote the vector space of all polynomials on R. Show that P is infinite- dimensional.

A: Let ℙ denote the vector space of all polynomials on ℝ .We have to show that ℙ is infinite…

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 X and Y be normed linear spaces and F : X → Y be linear. Prove that F is continuous if and only…

A: Let X and Y be matric space and let f: x→y be a uniformly continuous function. If ( xn)n∈N is a…

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: 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: 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…

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Q: Let R be the ring of continuous functions from R to R. Prove that M = {f e R : f(0) = 0} is a…

A: To show that the given subset M of R is a maximal ideal, we first show that M is a ideal of R and…