Let T be a linear operator on an inner product space V, and suppose that ||T(x)||= ||x|| for all x. Prove that T is one-to-one.

Q: Show that A is invertible if and only if AT is invertible.

A: Concept: Where the operator, operating on a function, produces a constant times the function, is…

Q: (d) If S,T are symmetric linear operators on an inner product space V, then their composition ST is…

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Q: Prove that if T is a unitary operator on a finite-dimensional inner product space V, then T has a…

A: Since, T is a unitary operator, there exists an orthonormal basis consisting of eigen vectors of T…

Q: _I. Let H and K be Hilbert spaces and A E B(H, K). Show that A is compact if = and only if A* A is…

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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: Let X be a normed linear space and T: X→ X and S: X → X be two bounded linear operators, then SoT…

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Q: 15. If T is a self-adjoint linear operator on a finite-dimensional inner product space V, then detT…

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Q: (12) Let T be a diagonalizable linear operator on V-finite dimensional, and let m be any positive…

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Q: If H is a Hilbert space and A € BL (H) is positive, prove that A+21 is invertible.

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Q: then L is a linear operatorE Let L(x): R R be a mapping such that L Select one: True False

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Q: If (T„) is a sequence of bounded linear operators on a Hilbert space and T, T, show that T,* T*.

A: Given Tn is a sequence of bounded linear operator on a Hilbert space and Tn→T We have to show that…

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: Prove that if T is a unitary operator on a finite-dimensional inner product space, then T has a…

A: It is given that T is a unitary operator on a finite dimensional inner product space.

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: Let P denote the vector space of polynomials (of any degree) and E P → R lenote the evaluation map…

A: Please find the solution in next step

Q: ITA Is a bounded self-adjoint operator on a Hilbert space н, show that |= sup {|}: xe H |x| = 1}

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Q: then L is a linear operator. Let L(x). R R be a mapping such that L( X2 Select one: O True O False

A: We will check whether the given statement is true or false.

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: Let E/F be a finite Galois extension. Let Kị and K2 be intermediate fields

A: Consider the equation

Q: Describe all linear operators T on R2 such that T is diagonalizable and T3 - 272 +T= To.

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Q: Let T be a self-adjoint operator on a Hilbert space H. Show that its norm is given by ||T|| = sup…

A: Given that T is a self-adjoint operator on a Hilbert space H. To prove: T=supx=1Tx,x.

Q: Let F = Q(V3+ V5). a. Show that F = Q(V3, V5). b. List all the proper subfields of F containing Q.…

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Q: 3. Define a mapping T: P3-P3 by. T(f(x))=f(x) Please find the nullity of T. (Hint: Still dx by using…

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Q: Let L be a linear operator on R1 and let a = L (1). Show that L (x) = ax for all x ∈ R1.

A: It's a highly simple question.

Q: Let A€BL(H) be a compact operator, then show that A*A is a compact

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Q: Show that if a non-zero element in a normed space X, then there is a bounded linear functional f on…

A: We need to show that if a non-zero element in a normed space X, then there is a bounded linear…

Q: LetT be a linear operator on an inner product space V. Let U1 = T + T*and U2 = TT*. Prove that U1 =…

A: T is an operator in an inner product space V over the field F together with an inner product. V×V→F…

Q: Let V be a finite-dimensional inner product space over F. (a) Prove that the trace of every positive…

A: Below given clear explanation

Q: Let L(x): R² → R be a mapping such that L X2 CH3 X2 then L is a linear operator. Select one: O True…

A: The given linear mapping is: L(x): ℝ2→ℝ2    such that:  Lx1x2=x2x1

Q: Let A be an operator on a Hilbert space H. Show that A is normal if and only if ||Ax|| =||A*x|| for…

A: Suppose A is a normal operator. Then by definition,  AA*=A*A            1 Now, consider  Ax2=Ax, Ax…

Q: 3. Let A, B, T be bounded self-adjoint linear operators on a complex Hilbert space H. If T20 and…

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Q: 6. Recall that G is a two dimensional space equipped with the norm I| (1, x2) ||2= V#1|² + \x2/?.…

A: Given: An operator A on l22 space is defined as, Ax1, x2=x1+x2, x2 To prove: a) A is linear…

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: 8. Suppose T: V V is a linear operator on an inner product space V. If T has an adjoint (hence, a…

A: Given that T is a linear operator on an inner product space V. If T has an adjoint. The unique…

Q: im(T1 + 14T) im(14T† + T2).

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Q: Let X be a pre-Hilbert space then [(x, y)|

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Q: Let T be a linear operator on a finite-dimenstona! inner product space V. Let E1,..., Ex be linear…

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Q: let P be the linear space of all polynomials of degree A. 1 B.n-1 C.n+1 D. n. E. None of the above

A: Let Pn the linear space of all polynomials degree less than or equal to n .

Q: Let T: H- » H be a bounded positive self-adjoint linear operator on a complex Hilbert space. Using…

A: As per the question we have to show that the Cauchy-Schwarz inequality holds for the Hilbert space H…

Q: Let V be an inner product space and T be a linear operator on V with || Tv||= ||v||  ∀ v ∈ V. Show…

A: Let V be an inner product space and T be a linear operator on V with Tv=v ∀v∈V.

Q: Exercise 3. Prove that a relation R on A is antisymmetric if and only if RNR-' C {(a, a) | a E A

A: We know that , A relation R on set A is antisymmetric  if either x , y∉ R  or (y , x) ∉ R whenever…

Q: Let f map C2 into C2 be a linear operator such that for every x in C2, f(f(x)) = x. Let V+ = {x in…

A: Let f:C2→C2 be a linear operator such that for every x∈C2, ffx=x.  Suppose V+=x∈C2:fx=x and…

Q: Let F be a field, and consider the linear map T: F5 → F5 defined by T(x1, x2, x3, X4, X5) = (0, x1,…

A: The objective is to find a basis of range T ,null T, and the dimension of range T and null T

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: Let X be a Hilbert space and T:X → X and T2: X → X are self-adjoint operators such that either…

A: Given we have given that X is a Hilbert space and T1 : X→X & T2 :X→X  are self -adjoint…

Q: Find examples of functions f and g such that f°g is a bijection, but g is not onto and f is not…

A: A function is said to be bijective if it is one-one and onto. We need to find f and g such that…

Q: Let A be an operator on a Hilbert space H. Show that A is normal if and only if ||Ax|| = ||A*x|| for…

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Q: Let V be a finite dimensional inner product space and T a non negative operator on V. Then T has a…

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Q: Let T be a self-adjoint operator on a finite-dimensional inner product space V. Prove that for all x…

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Q: (b) If f is a bounded linear functional on a Hilbert space H, show that there is a unique y e H such…

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