Let X and Y be normed linear spaces, and T E B(X,Y). Prove that 38 > 0 such that 8||x|| < ||T(æ)||, Vx E X 3T-1 E B(Ran(T), X).

Q: Let X be a normed space and aɛ X. Show that there exists an fe X' such that ||a|| = Sup { \f(a)| :…

A: Given that,                X is a normed space and a ∈ X . We have to show that there exists an f ∈…

Q: Let z=x+iy ∈ C where x∈R\{1}, y∈R, and |z|=1. If w = (z+1)/(z-1), then show that Im(w) = y/(x-1).

A: Use the following formulae, to obtain the required result. Real part of a+ib=aImaginary part of…

Q: Find a norm of the function f(t) = t² in space C" 0, 1, where norm is given by formula max|f(t)|+…

A: Solution

Q: c) For x e ll n2, define || x ]| = || x || 1+ || x ||2: Show that (11 n2, ||· || is a Banach space.

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Q: X= R² and T = { $₁,x} V {₂\x>0} where U₁ = {(x, y) | x² + y ² < 2² } Prove (XT) is Topology

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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 A={x e R|-1sx s1} = B, Show that f:A →B given by f (x)=x|x| is a bijecti on.

A: we have to show that given function is bijective

Q: In each case find the norm of the linear map L directly from the definition: (a) L : R → R with L(x)…

A: (a)  Given that the linear transformation L:ℝ→ℝ   with  Lx=-2x. The norm of the linear…

Q: Q3 Prove : If V is a normed linear space, and WcV, and U a closed subset of W Then given ve V,3u"…

A: given if V is a normed linear space, W⊂V and U is a closed subset of W then for v∈V there exist u*∈V…

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: 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 that P1(t), p2(t), and p3(t) are continuous on R, and that y1, Y2, and y3 are solutions to…

A: Wronskian of the function

Q: Let V and W be vector spaces, let T : V --&gt; W be linear, and let { w1 , w2 , ... , wk} be a…

A: Given:          V and W are vector spaces and T : V → W be linear.          The set w1, w2, ..., wk…

Q: = 0,1] and d be the usual metric, d(x, y) = a -y %3D en d can not induces a norm on X True False

A: the usual metric d(x,y)=|x-y| usually induces the norm.

Q: (v) Prove that if the space (X,T) has the fixed-point property and (Y, T1) is a space homeomorphic…

A: Given: v) To prove the given statement as follows,

Q: Evaluat..f.x.dA. wha.is.the..sgiem.in.the.fiigt. R quandrant..bounded..xy=16. X..8....y.z.. y.e.x...

A:

Q: Prove that if a e R, then a 0 = 0 = 0 · a. Suppose x, y, z E R. Show that xz = yz if and only if…

A: We will prove the given statements.

Q: * Let A E R"Xn, ĩ E R" \ {0} and A = 0. Prove that if the origin is stable if and only if ī is…

A: Given A∈ℝn×n,x¯∈ℝn\0 and Ax¯=0 To prove that the origin is stable if and only if x¯ is stable.  If…

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: 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: Suppose V is a vector space and S, T = L(V, V) are such that Range Scker T=null T. Prove that (ST)²…

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Q: Let V R2, W1 = {(x,0) : x E R}, W2 = {(0, y) : y € R}, and W3 Prove that V =W1 W2 = Wı W2 = W2 W3.…

A:

Q: If M and N are metric spaces, the Cartesian product M x N is again a netric space with the distance…

A: We have given that , M and N are metric spaces. We have to show that , M × N is again a metric space…

Q: Let T : V --&gt; W be linear, b ϵ W, and K = {x ϵ V: T (x) = b} be nonempty. Prove that if s ϵ K,…

A: Given: Let T : V --&gt; W be linear, b ϵ W, and K = {x ϵ V: T (x) = b} be nonempty The objective is…

Q: Given any Hermitian space E with › E, if (f(x), x) = 0 for all x € E, th I

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Q: Let X be a normed space and A € BL(X) be of finite rank. Then σe (A) = σ₁(A) = o(A).

A: 1

Q: 11. Let (V.||-|) be a normed vector space. Suppose e : V - R has the property that e(r+y) =…

A:

Q: i) Pe(x) = 0 iff r € Ë. ii) Prove that pe is uniformly continuous.

A: i Let E is the non-empty subset of a metric space X,d the distance from x to E is defined as…

Q: Consider the bounded linear operator T:C0,1]C0, 1] defined by -Ta =y y(t) = ta(t). Enter the norm of…

A: Given that T:C0,1→C0,1 is a bounded linear operator. Defined by x↦Tx where x is continuous function…

Q: Let V and W be vector spaces, let T: V → W be linear, and let {w1, w2, . . . , wk} be a linearly…

A:

Q: 1: Let (X,) be a metric space and let d1(x.,y)=K d(x.y), K>0. Prove that d1 is a metric function on…

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Q: If f(x) and g(x) have a period p, show that h(x)3Da f(x) + b g(x) (a, b constants) has the period p.…

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Q: Let V be an inner product space, and let y, z ∈V. Define T: V →V by T(x) = &lt;x, y&gt;z for all x…

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Q: Let v,w ɛ R" and define A = vw². Prove that ||A||2 result true for the Frobenius norm, i.e. is…

A: Let v,w∈ℝn define A=vwT. v=x1,x2,...,xn and w=y1,y2...,yn. Then,…

Q: suppose that x,y,z∈R3 are such that x.(y×z)≠0. prove that x,y,z are linearly independent.

A:

Q: Given a function f(x) = f(x1, x2) = (x2 − x 2 1 )(x2 − 2x 2 1 ), where x ∈ S =  [x1, x2] T : x1 ∈…

A: Given a function fx=fx1, x2=x2-x12x2-2x13, where x∈S=x1, x2T: x1∈ℝ, x2&lt;0. (a) Find out the nature…

Q: Find a norm of the function f(t) = 2t4 – tin space C0, 1], where norm is given by formula max|f(t)|…

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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 a function d: R2 x R2 R defined by d(x, y) = ((x1 – y1)² + (x2 – y2)²)* %3D = XA Vx =…

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Q: 12. Show that d((x1, Y1), (x2, Y2)) = |x1x2- Y1Y2| is not a metric on X = R %3D %3D

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Q: For normal linear spaces X, Y, prove that BL (X, Y) is complete if Y is complete.

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Q: Suppose that the set A={X₁, X2..., Xn}, X₁ ER" spans the entire space R", i.e. for any arbitrary…

A: Given: The set A=x1,x2,...,xn, xi∈ℝn spans the entire space ℝn, i.e. for any arbitrary n-vector…

Q: Find a norm of the operator A(x) = (t³ + 2t2)x(t) in space C[0, 1]. O a. 3 O b. 4 7 2 3 O d. 6 O e.…

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Q: if x is a compact Metric space and F e C (X₂Y) is equicontinuous, then F i's uniformly…

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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: Consider E = C( [o,(]) as normed spaces with normS II f | = sup 1fcxl and l fll,=Sifwildx. %3D Show…

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