Let S:x→Y be bounded lineay a operator, Then prove that ker S is closed.

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: Let T be function from R to R defined as T (z) = (r, x), %3D Then T is not linear operator O True…

A: FALSE  the detailed explanation is as follows below:

Q: Let r be a differentiable vector function of t. Show that if r . (dr/dt) = 0 for all t, then | r |…

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Q: Let ⁄2 be a bounded interval. For any g € L²(2), define the operator T, by T,(f) = f Then T, is a…

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Q: \Let X be Hilbert space over field F, and let R, S, T = S(X), 2 > 0: If S≤T then S+R≤T+R. (2) If S≤T…

A: Given : Let X be a Hilbert space over field F Let R,S,T∈SX , λ>0

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: be a linear operator on a vector space V(F). If = 0, t ce oft and null space of t. Also given an…

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Q: Define an operator T: fºº → f° by T(5₁) = (¹) # and bounded. for each (₁) E. Show that T is linear…

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Q: Derive a Dirichlet principle for the elliptic operator L in divergence form with A e C² and b= 0.

A: Given elliptical operator L~ in divergence form with A∈C1 and b=0. Suppose A∈C1Ω of the second order…

Q: normed lineer spare X is finite demisional Linear transformation. X is bounded om If them every

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Q: Use the definition of the dot product a b = |a||b| cos 0 to prove the Cauchy-Schwarz Inequality: |a…

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Q: Prove that if r is a vector-valued function that is continuous at c, then || r || is continuous at…

A: Consider the vector r→a=p(a)i+q(a)j+s(a)k.   If r is continuous at a=c. So, lima→cr→(t)=r→(c) Then…

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: Let fogec[agb] where c [a, b] Lt is the Vector sPace of the continuous functions overthe…

A: Let , f , g ∈ Ca , b  , here Ca , b is a vector space of the continuous functions over the closed…

Q: Show that with the subspace topology Uy on Y, the inclusion map : Y → X is continuous.

A: Given that, UY is subspace topology on Y. Suppose (X, Tx) be topological space and Y is subset of X.…

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 X-j m,n; is Z Ž asigma algebra over X ?

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Q: Let V be a vector space over R, and dimV=200. Let NEL(V) be a nilpotent operator such that null(N2)…

A: Consider the given information.

Q: Evaluate f(x²y+3xyz) dxdydz by applying the transformation u = x, v=xy and w=3z, where G is G region…

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Q: 9. Show that a bounded linear operator T: H–→H on a Hilbert space H has a finite dimensional range…

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Q: a) Define a linear operator on R' by performing the following operations in order: i) a shear in the…

A: We are authorized to answer three subparts at a time, since you have not mentioned which part you…

Q: Prove that f, uv v • dỉ = – §̟ vũu • dĩ

A: Whenever the force is conservative in nature the work done over the closed loop is always zero. i.e…

Q: Is = xjx + yjy+zźz3 an inner product on R'?

A: No it is not an inner product.  Inner product looks like a dot product.

Q: Let X = Rand de f ine ||| on X as ||r|| = Va then X is normed linear space

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Q: • Exercise 3 Prove that if S and T are both bounded linear operators from the normed space X to the…

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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: If M = {x: Lx = 0}, where L is a continuous linear functional on H, prove that M¹ is a vector space…

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Q: #Let Vz arih Here, れgl。 and hahe what netationship must hold betaen adho foro v to be ewoatly…

A: The expression for the volume is V=πr2h.   Thus, the total differential is given by dV=∂V∂rdr…

Q: Let VER, defined the addition by: %31 with standard multplication - IS V Vector sPace Just

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Let T be the linear functional defined on C [a b] (with sup norm) by T (x) = ex (t) dt for x eC[a,…

A: Since T is a linear functional defined on        C [a,b]

Q: If F is a conservative vector field on a region R, how do you evaluate∫C F ⋅ dr, where C is a path…

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Q: a bonneled linear operator. Then prve. that. Kes's is clased. b) let 8:X>y be a

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Q: LetT be a normal operator on a finite-dimensional inner product space V. Prove that N(T) = N(T*) and…

A: Let T be a normal operator on a finite-dimensional inner product space V

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: 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: Let T be function from R to R2 de fined as T (x) = (0, x), Then show that if T is linear operator or…

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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: On a vector space W of all continuous functions from R to R, define I, : W → W as I,[f(t)]= J', f(t)…

A: Yes  given function is linear transformation

Q: Let T be a linear operator on a finite-dimensional Prove that T is normal if and only if omplex…

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Q: On a vector space W of all continuous functions from R to R, define I, : W – W as I[f(t))= J', f(7)d…

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Q: Let T be a linear operator on a finite-dimensional Prove that T is normal if and only if eomplex…

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Q: ƏF dF = (dr. V)F+ -dt. at

A: Given: A vector function F depends on both space coordinates x, y, z and time t.  To Find:  To show…

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: Consider the operator T: 1 defined by a = whether the following statements are true or false. (1,…

A: In Codomain only first term of domain will be exclude.  Hence,  null space will be (a, 0,0,.....)…

Q: Let F(x, y, z) =⟨3x2y+az, x3,3x+ 3z2⟩ be a vector field. For what values of a is F conservative?

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Q: Let T be function from R to R² de fined as T (x) = (x, 0), Then show that if T is linear operator or…

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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: Suppose a vector field F(x, y) = wai + VEj is conservative. Then construct a potential function ø(x,…

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Q: Determine whether the following vector field is conservative on R. F- (x.4y)

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