Prove that the map || - || : L(V, W)→R defined by ||L|| = sup {|| L(x)|| | ||| < 1} is a norm on L(V, W).

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: (Proving Linear Algebra)

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Q: Explain why the Lp[0, 1] spaces are complete normed linear spaces. What is the norn of function f(x)…

A: This is a problem of functional analysis.

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: 10. Prove that T, is topologically conjugate to the quadratic map F(z) = 4z(1 - z).

A: Solution of 10. To prove that T2 is topologically conjugate to the quadratic map F4, we have to find…

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: Suppose V₁, Vm, and W are vector spaces. Prove that L(V₁ × ... X Vm, W) are isomorphic to L(V₁, W) ×…

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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: 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: Let V, W be finite-dimensional inner product spaces and T € L(V, W). Prove: (a) If T is injective,…

A: Injective Function:  A function is said to be injective if each element of the codomain is mapped to…

Q: Let T be self-adjoint on a finite-dimensional inner product space V. Prove that Vr E V, ||T(x)± i||?…

A: The complete solutions are given below

Q: 2. If X,Y are Banach spaces on XOY define |(x, y) = |, +|yly . Show that this gives a norm making X…

A: Given: Dual of c0  is isometric to l1 State the principle of uniform boundedness.

Q: Theorem 2.9. Suppose p ¢ A in a topological space (X,T). Then p is not a limit point of A if and…

A: Given that, Let (X, T) be the topological space. p is the not element of set A. We have to show…

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 the multiplication map · : ℤ/n × ℤ/n → ℤ/n given by [x] · [y] = [xy] is well-defined.

A: A multiplication operation defined on ℤ/nℤ×ℤ/nℤ→ℤ/nℤ as                x·y=xy

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: 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: Show that the maximum norm on C[a, b] is not induced by an inner product

A: to show that the maximum norm on Ca,b is not induced by an inner product . as we know , a norm X,·…

Q: Let T be an invertible linearlo-perator on a finite-dimensionalvector spaceProve that T is…

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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 &lt;x, z&gt;= 0 for…

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

Q: 3) Prove that the multiplication map · : Z/nx Z/n → Z /n given by [x] · [y] = [xy] is well-defined.

A: To prove that the multiplication map · : ℤn×ℤn→ℤn given by    x·y=xy  is well-defined.

Q: Where x = (X1, X2) E R². 3- N: R2 - → R defined by N(x) = /x1² + x2² is a norm on R². (x1, x2) E R².…

A: Let X=(x1,X2)€R^2

Q: Find a norm for C([0, 1]), so that C([0, 1]) is not complete under this norm.

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Q: Let X be a finite dimensional normed space (a) and A :X→X is a linear map. Show that A is compact.…

A: Given: X is a finite dimensional normed space and A : X → X is a linear map. We have to show that A…

Q: Theorem 7.29. Suppose f : X → Y is a continuous bijection where X is compact and Y is Hausdorff.…

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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: Prove that ℝ with lower limit topology is Hausdorff

A: Solution :-

Q: Let (X, d) be a compact metric space and let A ⊆ C(X; C) = {f : X → C; f is continuous} be an…

A: please see the next step for solution

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: Let X and Y be topological spaces. Prove that X x Y is completely regular if and only if X and Y are…

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Q: (b) Prove that the unit ball of a normed linear space is compact if and only if the normed linear…

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

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Q: Prove that Let X be a metric space such that X E1 U E2. If E1, E2 are complete, then X is complete.

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Q: Theorem 2: Let X be a finite dimensional normed space and let r > 0. Then, the closed unit ball B[0;…

A: Theorems and definitions we are going to use: 1. Definition: (X,|| . ||) is said to be bounded if…

Q: (b) {T } is a sequence of bounded linear maps on a Banach space X and Tx = lim 1x exists for all xE…

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Q: (a) If X,Y are Banach spaces on XOY _define |(x, y)|| = || +|>ly. Show that this gives a norm making…

A: If X and Y both are Banach spaces then we need to prove that X⊕Y is Banach space with respect with…

Q: Show that if a contraction map f: from M to M has a fixed point then it's unique.

A: Definition: Let X, d be a metric space. The map T: X→X is said to be a contraction mapping if there…

Q: 2.9.11 Prove that every function f: Z→ F is continuous, where the metric on Z is the restriction of…

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Q: 14. If d is a metric on a vector space X‡ {0} which is obtained from a norm, and å is defined by…

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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: Define A : L3/2[–1, 1] → C by Prove that A is a bounded linear map and compute its norm.

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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: 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…

Q: b) Let f(x) = 1 and g(x) = x be functions in C[0, 1]. Use the inner product defined in Example 5, to…

A: Note: Since there is no proper instruction is given and the first part is incomplete, we have solved…

Q: Letf(z) 3z-2i. Find the maximum valne of /(=} on the disk z<.

A: Triangle inequality:z1-z2≤z1+z2 and z1+z2≤z1+z2

Q: If x is a normed linear space and the closed Unit ball M is compact, then X is finite dimensional.

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Q: Let X# and t1, t2 are two topologies on X such that 11CT2. Prove or disprove that if (X,t2) is a…

A: Counter Example, The set of real numbers with usual topology. That is,  ℝ,τS usual topology space.…