11. Let (V.||-|) be a normed vector space. Suppose e : V - R has the property that e(r+y) = 2r{r)e(s) for every r.y E l. Prove that if e is continuous at 0, then e is continuous at every r E V.

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: 11. LetF(x, y) = (ye™ – 6x))i + (xe"» + 4)j %3D (a) Show that F is a conservative vector field. (b)…

A: we need to show F conservative vector field then  potential function f=

Q: Let V be a vector space over a field F and let w, x, y, z ∈ V. Show that w + (x + (y + z)) = ((w +…

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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 n…

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Q: 3. Prove that if r, y E RP, then ||r + yl|2 = ||x||2 + |ly||² holds if and only if r y = 0. 4. Prove…

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Q: Suppose Y is a closed subspace of a normed linear space X and xo ¤ Y, x, € X . Prove that there is a…

A: Y is a closed subspace of a normed linear space X and x0∉Y, x0∈X.. We have to prove that there is a…

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: 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: Let V be a finite-dimensional vector space over F, and let S,T € L(V) be linear operators on V with…

A: Consider the provided question,

Q: 5) Let V, W be finite dimensional vector spaces over F. (a) ( Prove that (b) Use this to prove that…

A: Question is wrong • please see the justification

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: Suppose that F is a force field defined in a set D c R? Simply related. If for every simple closed…

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Q: Consider E =IR as a vector space. Show that: A function f: E → IR is a norm if and only if fu) =klxı…

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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: (6) Suppose that T is a linear operator on a 2 dimensional vector space V and that T+ a I for any a…

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Q: 4. Let F be a smooth vector field. Show that the flux of F leaving an infinitesimal cube of volume…

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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: Suppose that W is a finite-dimensional vector space over F and that T:V→W is linear. Prove that…

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Q: Let X and Y be normed linear spaces, and T E B(X,Y). Prove that 38 > 0 such that 8||x|| < ||T(æ)||,…

A: Consider the given information.

Q: If a vector field F (x, y, z) = (y² + z², xe cos z, - xe³ cos z), which of the following values is…

A: Given vector field is Fx, y, z=&lt;y2+z2, xey cosz, -xey cosz&gt;.   To Find: div F.

Q: 8. If F is a conservative vector field and S is a sphere of radius a enclosing a region D, then…

A: For any conservative vector field F, the integral over any curve C, is dependent on the starting and…

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

A: Given : V is any normed vector space and x is any non zero vector and α&gt;0. To find :  A real…

Q: 4) Determine whether or not F is a conservative vector field. If it is, find a function f such that…

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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: (d) Consider the vector space C|0, 1] of all continuously differentiable functions defined on closed…

A: We have solve the integration

Q: Suppose V is a 4-dimensional vector space, and that T is a linear operator on V with minimal…

A: Dimension of Ker T

Q: Show that if the vector field F = Pi + Qj + Rk is conservative and P, Q, R have continuous…

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Q: Let V and W be vector spaces and T:V→W be linear. Let {y1,…,yk} be a linearly independent subset of…

A: It is given that Let V and W be vector spaces and T: V→W be linear. Let {y1… yk} be a linearly…

Q: Suppose S is the surface where x² + y² + 3z² = 1. Let F be the vector field F = (xy, yz, z²).…

A: We need to evaluate integral.

Q: 32. In a normed vector space (V, ||) show that B,(x) = x + B,(0) = {x + y: Ilyll <r} and that B,(0)…

A: Definition : Let ( V , || . || ) be a normed vector space .For any r &gt; 0 ,  the set Br (x)={ y ∈V…

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: 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 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: Let V and W be finite-dimensional vector spaces and T: V →W be linear. (a) Prove that if dim(V)…

A: (a)Let B={v1,…,vn} be a basis for V. Then, C={T(v1),….,T(vn)} spans T(v)

Q: c) Prove that a subset W of R’ defined by W ={(x, y, z) e R² / 2x= y = 2z} is a vector space of R'.

A: Given, W = { (x,y,z) in R³ | 2x = y = 2z } is a subset of R³ Let, u = (x,y,z) &amp; v = (a,b,c)…

Q: 5. Let V be a vector space, and let x1, X2,.. . , Xk E V. 5(a) Prove that if either x1 = 0 or x;+1 E…

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Q: Consider the linear space C[0, 1] equipped with the norm ||f || = max |f(t)|. te[0,1] Prove that…

A: Given: The linear space C0,1 equipped with the norm, f=maxt∈0,1ft   To prove: There is no inner…

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: Let V be a finite-dimensional vector space, and let T: V →V be linear. (a) If rank(T) = rank(T2),…

A: Part (a): Let Dim V = n then the dimension theorem applied to the transformation T states that rank…

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: A Consider E = \R as a vector space. Show that: if and only if fixed real winber A function f: E →…

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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: Let X is a normed linear space and the ball M= {xe X:|| x ||≤1} is compact, then X is finite…

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Q: 10. Suppose that V is a normed vector space over F. Let S = {x € V : ||x|| 1 is not an accumulation…

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