Q If X R, define the function ||- ||: R →R*u {o} by ||-|| = |x| + 1. Is (R, ||-||) normed space. Explain. %3D

Q: ) Consider the sequen ces equi' pped space f absolutely ce uith He Ofalloing' onvergent norm (0.1)…

A: We will use the basic knowledge of norms and functional analysis to answer all three parts of this…

Q: Let the ordeced two bases of the spaces B be A=- (1,3) (E),L)} and B=fF3,2).(5,1)} ln the case, find…

A: We required to find the transition matrix from base B to base A Which is given follow

Q: llnll=max |x C+)| tej show that is norme space. j-[a,b]

A: We have to show that x=maxt∈jxt, where j∈a, b, is a normed space. For this, we have to use the…

Q: (b) Is (R2,d) a metric space, where d((x.y1).(x2.y2)) 1x1-x2l; for all (x, , y1 ), (x2 2) E R2 ?…

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Q: Ques. 1. Let N be a linear space and d be a metric on N such that d(x + z,y+ z) = d(x,y), x,y € N…

A: I have proved all three conditions gor norm

Q: Q2:(a) In Euclidean metric space (R, 1. 1),if A = {y € R: y = 2 cos (2x); x € R} then fined the…

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Q: Consider R2 equipped with the Euclidean metric d2. Let A be a non-empty subset of R2 , that is…

A: Consider R2 equipped with the Euclidean metric d2. Let A be a non-empty subset of R2 , that is…

Q: f If Xo's Compact, x=y Continuous bijective L Topelagied space Then f is homeomorphism

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Q: Let X = Rand def ine |-|| on X as ||r||= V then X is not normed linear space O True False

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Q: 3. Let Ac R' where A = {x: x 1}. Describe the quotient space topology on R'/A obtained by…

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Q: A normed space X is said to be strictly convex if for x # y in X with |||| = 1 = ||yl|, we have…

A: It is given that X is normed space and for x≠y in X with x=y= 1, then  x+y<2. We have to explain…

Q: Examples 1) X = R with the norm |x|| = |x| x ER is normed space %3D

A: We need to show set of real number forms a normed linear space under usual modules operation.

Q: A metric space (X,d) such that the function d1(p, q) = (d(p, q)) i not a metric in X

A: Since (X, d) is a metric space. So, d is a function defined as d : X×X→R+∪0 and satisfy the…

Q: Let X = Rand def ine ||.|| on X as ||x|| = 1 %3D then X is not normed linear space True O False O

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Q: Let X be a normed space then 1- ||0|| = 0 2-I1-xl = |지l, Vx E X 3- ||x - yl| = |ly - x|,Vx, y E X…

A: As per the policy, we are solving first three subparts. Please resubmit the question and specify…

Q: (b) Show that x. y < ||x|| ||y||- [Hint: If x, y # 0, let a = , 6= and use the fact that ||ax + by |…

A: Given that, Euclidean metric d on Rn is defined as d(x,y) = || x - y || for x, y in Rn Part(a): Let…

Q: Let X = R and de f ine ||.|| on X as ||a|| = Va2 %3D then X is normed linear space O True O False

A: The given norm is Euclidean norm.

Q: Q4. Consider the topological space (X,T), where X = {a, b, c, d) and T = {o, X, {a}, {b}, {c},…

A: Given: Q4. a)  X, T is a topological space, where X=a, b, c, d and T=∅, X, a, b, c, {a, b}, {a, c},…

Q: Q3 Assume that (M,d) is a Compact metric Space. Show that, f: (M,d) → (Y, 2) is Continous bjective…

A: f:(X, d) to (Y, d) is said to be homeomorphism if  1) f is continuous.  2) f is bijective.  3) f…

Q: (a) If ker(L) has dimension s, what is the rank of L? (b) If ker(L) # {0v}, show that L is not…

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

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Q: Find all valus of z such that (5 + z)6 = (1+ i)z6

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Q: Give an example of (v1, v2, v3) and (w1,w2, w3) independent se- quences in the Rº such that…

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Q: Exercise. Prove that in any metric space (X, p), a closed ball {r E X : p(a, x) <r} is closed.

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Q: frnd an o-thogonal besis for the {(0,2,1),,0,1), (lo,4.0), (4,2,2){ space spanned by dac set

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Q: roblem 3 Let (X, M,µ) be a space with measure. Let and A,B E M µ (B)=0. Prove that

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Q: Q2: Let (R,7) be the usual topological space f: (R,r) (R,7) fx? +1 :x 21 st. Ar) =-x ;x<1 . Isf…

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Q: xamples -X = R with the norm ||x|| = |x| x € R is normed space

A: Here we needs to show real number is a normed space under modules.

Q: (13) Show that | l is a norm on the space L [a,b].

A: Please check step 2 for the solution.

Q: Q2: (a) In Euclidean metric space (R, 1. 1),if A = {y € R: y = 2 cos (2x); x € R} then fined the…

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Q: Exercise.3 Show that the following the functions d and p on X = R? are metrics or not, such that 2.…

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Q: c spaces. Show that X x Y and Y

A: The sup-metric d1 on X×Y is defined as d1x1, y1, x2, y2=max dx1, x2, ey1, y2. Similarly, the…

Q: z? f(2) = 1-z* 4

A: We use the fact that the 1/(1-z)= 1+z+z2+z3+z4+......

Q: 2. Let (x, d) be that the following a metric space, ond ACX. Show equivalent. i A is closed ii. A=…

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Q: Q4 Let V=R, defined the addition by: %3D with slandard scalar multiplication. Is Va valtov space…

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Q: Given the expression |z + 4 - li|S Bnd the geometric region of space where the z exists in. Graph…

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Q: Q4. (a). Let V be an inner product space. Show that Il u+ v ||? + || u – v ||= 2(|| u ||° + || v…

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Q: (b) Let X bę a normed linear space. Let Вх, г) %3D (у є Х: || x-у || <r). Show that B(x, r) = {y e…

A: To show - The ball B¯(x, r) is closed. Indeed, for any given y ∈ M \ B¯(x, r) we set R = d(x, y)…

Q: Ques. 1. Let N be a linear space and d be a metric on N such that d(x + z,y+z) = d(x,y), x,y € N and…

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Q: Q5. Let (X, B(R), µ) be a Borel measure space. Suppose that {A, = ["+1_ 4n-1 Show that A1 C A2 C…

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Q: a) Find a space that is a regular but not Hausdorff. Justify your claim.

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

Q: (b) space and X+{ 0 }. Show that B< (X, Y) is a Banach space. Let X be a normed space and Y be a…

A: X be a normed space and Y be a Banach space We have to show that B(X, Y) - set of all boundes linear…

Q: (N) The mapping ||x||(--) := V(x|x) : X → R is a norm on X. CS) |(x\y)| < ||||(1) ||y||41» (x, y € X…

A: We shall solve first three subpart only asper the company guideline- for other kindly post again by…

Q: Exercise. Prove that in any metric space (X, p), a closed ball {x E X : p(a, x) <r} is closed.

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Q: Ques. 1. Let N be a linear space and d be a metric on N such that d(x + z,y + z) = d(x, y), x,y E N…

A: Given: N be a linear space and dbe a metric on N such that d(x+z, y+z)=d(x, y),  x, y ∈ N and  d(ax,…

Q: Q3/ Choose the correct answer a, b, c, or d. 1) The compactness of a subset V of a normed space F is…

A: We will use basic knowledge from the theory of normed linear spaces to answer this question. Note…

Q: Qz\ show that ?_C=0} is not a swbopace of R

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Q: Q#1: Discuss Application of inner product spaces with examples.

A: Important Application of Inner product spaces is to find Least squares approximations for a…

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.