(b) Suppose -|, and ||-||2 are two equivalen norms on a vector space X. If (X, ||.||,) is complete, then show that so is (X, ||-||,).

Q: 1. Suppose that x is a non-zero vector in a normed vector space V over F, and that a > 0. Find a…

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Q: If V is a vector space then show that cV=V.

A: Vector space: A non-empty set V closed with two binary operations called addition and scalar…

Q: = {x = (r1, 12, 13, - bsolutely summable infinite sequ is a complex vector space.

A: Let (X,d) be a complete metric space with X not ∅. Suppose the function f : X → X has the property…

Q: Give an example with values of a nonstandard operation for M2*2 a vector space that fulfills all…

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Q: Is it possible to make Z into a vector space if we redefine addition?

A: To check: the possibility to make ℤ into a vector space.

Q: Every finite dimensional vector space has an orthogonanal basis. False O True

A:

Q: If (x1,y1) + (x2,y2) = (x1x2,y1y2) and  c(x1,y1)= (cx1,cy1) are vector spaces, verify if both…

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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: Prove that L∞[a, b] is complete with respect to the norm || · ||∞.

A: The elements of L∞[a,b] are the essentially bounded measurable functions.

Q: Q6: Show that R³ = {(x1,x2, x3)|lx1,x2, x3 E R} is a vector space

A: Vector space is the field created by set of vectors with specified direction and magnitude, usually…

Q: See picture

A: Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the question and…

Q: Can a vector in a vector space R -4 - 2 be 2. span by -1 2 イ

A: Span of vectors.

Q: 3/ Given a vector x, show that the infinity norm defined by || X 1100 = max | X₁ Do Ksign

A: Given a vector X A norm is a function such that the following conditions hold ∥x∥≥0 ∀x∈Rn ∥x∥=0 if…

Q: Show that the element 0 in a vector space is unique

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Q: 3. Prove that Ov=0 for every vector v in a vector space.

A: We have to prove that 0v = 0 for every vector v in a vector space.

Q: Theorem 3.1. Let V be an inner-product space and vectors x, y e V, then |(x, y)|< || x || | - Strict…

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Q: Prove that a vector space is infinite-dimensional if and only if it contains an infinite linearly…

A: Given the statement that a vector space is infinite-dimensional if and only if it contains an…

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: If scalar multiplication and addition is redefined on R² in the following way, show that it is NOT a…

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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: Every finite dimensional vector space is

A: We have to prove the given statement.

Q: Every finite dimensional vector space is algebraically reflexive. Prove that.

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

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Q: 12. Suppose that a, b e R and a < b. Show that the vector space C[a, b] of all continuous complex…

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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: (b) Show that the set of all positive real numbers, with x+y and cx redefined to equal the usual xy…

A: Here, to check that the set of all positive real numbers with  x+y=xy and cx=xc where xy is usual…

Q: w that the space V = {(x1, x2, x3) ∈F3

A: To prove that the space V=(x1, x2, x3)∈F3|x1+2x2+2x3=0 forms a vector space.

Q: Suppose that W1, W2,. , Wm and V are all finite dimensional vector spaces over F. Prove that 2,...…

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Q: 3. Prove that in any normed linear space, xo is the Chebyshev center of the closed ball {x : ||x –…

A: Given: Given any normed linear space x

Q: Which of the following are axioms of a vector space V? (Select all that are correct.) 1x = x If k,…

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Q: Let P2 be the vector space of all polynomials of degree ≤ 2 with coefficients in R, and S= {1 + 2x,…

A: The basis for the vector space P2 is 1, x, x2. We have to write the matrix form for the set S. We…

Q: Either use an appropriate theorem to show that the given set, W, is a vector space, or find a…

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Q: Find the dimension of the vector space.P4

A: Here the given vector space V= P4 or

Q: Show that the set of all pairs of real numbers (x, y) with the operations (X1, Y1) + (x2, Y2) = (x1…

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Q: prove that any finite dimensional normed space is reflexive.

A: Let us suppose that we have a finite-dimensional normed space, X For any natural number, n, let…

Q: 2. Prove or For finitely-generated vector spaces U, V, W disprove: r F, L(U, W) = L(V, W) if and…

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Q: Is it true that if two vector spaces V and W both have dimension 2022, then they must be isomorphic?…

A: Suppose V and W are two vector spaces. We know that , V and W are isomorphic to each other if there…

Q: The vector spaces M2, 2 and M5,1 are isomorphic to each other ylgn ihi

A: False

Q: 1 3 5 Find an explicit description of Nul A by listing vectors that span the null space.A =

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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: Suppose W1, ..., Wm are vector spaces. Prove that L(V, W1 × • × Wm) and L(V, W1) × ·…× L(V, Wm) are…

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Q: Only part d please.

A: Given:

Q: Prove that (Pn, +, ·) is a vector space.

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Q: If V (F) is a finite dimensional vector space of dimension n then each set of (n+ 1) or more vectors…

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Q: The set R2 is a vector space over R. How Prove it ?

A: We need to check each and every axiom of a vector space to know that it is in fact a vector…

Q: If V1, V2, V3, V4, and W are all vector spaces.  Are L(V1x...xV4, W) and L(V1, W) x...x L(V4, W)…

A: Given that: V1,V2,V3,V4,W=Vector space The objective is to verify the given statement. Using the…

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: Therefore (0,0, 0) e W. Ex 6. If W, and W2 be any two sub-spaces of a vector space V, then inW, is…

A: We have to prove that intersection of two subspace of V  is also a subspace of V

Q: 10. If A is 3 x 3 with rank A = 2, show that the dimension of the null space of A is 1.

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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.…