9. Show that a bounded linear operator T: H–→H on a Hilbert space H has a finite dimensional range if and only if T can be represented in the form Tx = 2 (x, v;)w; [v, w, e H]. j=1

Q: Calculate the norm of the linear functional 1 į t f(t) dt on each of the spaces 0 (C[0, 1], || ||)…

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Q: . Show that the operators T₁,, T4 from R2 into R2 defined by (1, 2) (1, 0) (§1, §2) (0, §₂) (§1, §2)…

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Q: Consider the right-shift on the Hilbert space l2: S:l → l, S(@1,a2, ...) = (0, a1, a2, ...). Define…

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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 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: 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: Let f(t) = (t – 2)*(t+1)² be the characteristic polynomial for a linear operator T. Obtain all…

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Q: 10. Let X be an inner product space and T: X→→→→→X an isometric linear operator. If dim X<∞, show…

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Q: 3. Let L be a linear operator on R and let L(1) = a . Show that L(x) = ax for all x E R .

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Q: Accept that (C'[0, 1], || - ||1) is a Banach space, where c'(0, 1] := {x € C[0, 1] : a'(t) E C[0,…

A: Consider the given C10,1:x∈C0,1:x'(t)∈C0,1 and x1=supt∈[0,1]x(t)+supt∈[0,1]x'(t)ddt=1

Q: 3.3.9. Given a normed space X, prove that the function p: X R defined by p(x) = ||x|| is continuous…

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Q: Suppose {u1, Span{u;}nSpan{w;}={0}. Ur, W1, Ws} is a linearly independent subset of V. Show that •..…

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Q: Show that the set {1, x, x2, ... , xn} is linearly independent in Pn(F).

A: Given, {1, x, x2, ... , xn}

Q: 6. Consider a normed vector space S = (R,|- |). Show that, for all r, y e S, |z - y| 2 | – l9|-

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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 linear operator defined by Show that T is compact. E1 E2 E3 T:l² l², x = (1, 2, 3, ...)…

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Q: et X be a normed linear space and T: X→ X and S: X→ X be two perators, then S T :X→X is a bounded…

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Q: 4. Show that for any bounded linear operator T on H, the operators T₁=1/√(T+T*) and T₁=1 (1 ==…

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Q: Let L:R R be a linear operator such that L X2 X3- X2 then Ker(L) spanned by: Select one: (e, e )

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Q: 7. Let T, and T, be bounded linear operators on a complex Hilbert space H into itself. If (T,x,…

A: Given: &lt;T1x, x&gt;=&lt;T2x, x&gt;, ∀ x ∈ H. To prove: T1 = T2.

Q: If S and T are normal linear operators satisfying ST* = T* S and TS* = S*T, show that their sum S+T…

A: I have used just definition of normal operator

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 H, and H, be Hilbert spaces and T: H H, a bounded linear operator. If M,cH, and M2 c H2 are such…

A: Given H1 and H2 be Hilbert space and T:H1→H2 a bounded linear operator. If M1⊂H1 and M2⊂H2 are such…

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

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Q: If T: H→→H is a bounded self-adjoint linear operator and T‡0, then T"#0. Prove this (a) for n = 2,…

A: Given T:H→H is a bounded self-adjoint linear operator and T≠0. We know that if T is a self adjoint…

Q: Let L: R2 → R2² be the linear operator defined by L 2

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Q: Let (x, Il.),4 11.11) be real Bonach spaces ond a bounded linear operator. Show a dense set in T(x)y…

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Q: 2. Use Wronskian to show that fi = x, f2 - Inz, and fa C (R). z Inz are linearly independent in

A: First we will calculate the wronskian of given functions f_1,f_2,f_3

Q: Let Te B(H) be an invertible operator. Show that a(T-) = 0(T)-.

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Q: .

A: Given: L is the linear operator on and it is define by,

Q: Let A: C [0,1]→C [0,1]. Prove that operator A is continuous reversible, find A. Ax() = x() + ja-m}xt…

A: Solution:- We have given operator A such that A: C[0,1]- C[0, 1] Also we must know that the inverse…

Q: If X is finite dimensional hor med space o T: X-Dy is a Linear operator from X to normed space y.…

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Q: 8. Show that an isometric linear operator T: H→→ H satisfies T*T= I, where I is the identity…

A: Let T : H → H  be an isometric linear operator. We need to show that , T satisfies T* T = I We know…

Q: Suppose T E L(U,V) and S € L(V, W) are both invertible linear maps. Prove that ST E L(U, W) is…

A: First we prove that ST is a invertible linear map from U to W and then it will be shown that the…

Q: Let A be a bounded linear operator on a Hilbert space H and ||A(x)||=||x|| ¥xe H. Show that A is…

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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: 1 Are the following sets of functions, each defined on x E R, linearly independent? Use Wronskian.…

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Q: 4) Let V be the vector space of polynomials of degree < 2 and let p(x) = ax² + bx + c. Find the…

A: Given that, The definition for adjoint linear operator is, Let V and U be two vector space over the…

Q: Show that the set {1, x, x2, ..., xn} is linearly independent in Pn(F).

A:

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: 1 The natural domain of the vector-valued function 7 (t)= In(In is O (-3, 3) O (1, 3) O R O (0, 3)…

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Q: Conversely, if T :V V is a linear operator such that T2 = T, show that V = kerT O imT. [Hint: 3.b, v…

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Q: 19: Find two linear operator T, and T,on V(R) such that T,T, = 0 but T,7, - 0.

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

A: If T is an operator in an inner product space V over the field F together with an inner product,…

Q: Give an example of a linear operator T on an inner product space V such that N(T) ≠ N(T∗).

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Q: Show that each linear operator T on R2 is nonsingular and find a formula for T- , where (a) T(x,y) =…

A: We have to show that linear operator T on R2 is nonsingular andfind the formula for…

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.

Q: Q3- Let S and T be bounded linear operators on a complex Hilbert space H into itself. If (Sx, x) =…

A: Let S and T be bounded linear operator on a complex Hilbert space H into itself. If…

Q: For a linear operator T on an inner product space V, prove that T∗T = T0 implies T = T0. Is the same…

A: Let T be a linear operator defined on an inner product space V.To prove