(a) Explain why X meets the criteria to be a discrete r.v., and give its space. (b) Find the pmf of X.

Q: If llall is a vector norm for a convex XEIR show that f(x)=11x11 is function.

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Q: How do you know that for all a, b, c, d: ax + bx will be orthogonal to cx? + d ?

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Q: c. Suppose that G,H < Sym(Q). Show that the corresponding spaces (2,G) and (Q,H) are equivalent if…

A: Suppose that G, H ≤ Sym (Ω). Let G, H be a multiplicative group and let Ω be a set. An action of G,…

Q: 2.4.5. Let X be a metric space. Given x E X, prove that {x} is a closed subset of X.

A: Let (X,d) be a metric space to show that {x} is closed sunset of X.

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: (a) Is the set S = {1 + x + x2, 1 − x3,  x + x2 + x3,  x3} ⊂ P3 linearly independent? If so, prove…

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Q: S ⊂ R3 be the set consisting of 2 points (1, 2, 2) and (1, 2, 3). Find the annihilator of S. Show…

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Q: Thm. Let v be an inner product Space then and HyE V жує ||≤|12||· 1191/ further equility is hold iff…

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Q: d) Consider the following equation of non-negative integers, x1, 12, T3 Ssuch that r1+12+ T3 = 20.…

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Q: 3. Let (X, 7) be a Hausdorff space. If X is finite, then show that 7 = P(X), where P(X) is a power…

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Q: Consider the set V = {(x, y) = R² : x+y=1}. on this set defined in the following way: (x2, y2) = (x₁…

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Q: the quotient space of a T0−space necessarily T0

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Q: For each of the following functions f(x) (x E R²), describe the sets V₁ ≥ 4 = { x | f(x) ≥ 4 } and…

A: A function is called convex if its double derivative is positive OR any line formed by joining any…

Q: Show the 3-dimensional ball B₁: ((..) ER³ | x² + y² +2² 0.

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Q: From the given theorem I need the other part of the proof of (c) since the first one is solved. I…

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Q: a) ||x|| = (x1)² + (x2)² (Euclidian norm) b) ||x|| = (x1)² + 5(x2)² c) ||x|| = |x4| + |x2|

A: As per the company rule, we are supposed to solve the first three subparts of a multi-parts problem.…

Q: Find the dimensions of the following linear spaces. (a) RSX3 (b) P6 (c) The real linear space C

A: In general Rm×n is a vector space of dimension m×n. So, Dimension of R5×3 is 5×3 = 15 (a) 15

Q: Suppose A is a function that assigns to each subset of the plane a number in such a way that the…

A: Given that AX≥0 and if X and Y are disjoints then AX∪Y=AX+AY To prove if X⊂Y then AX≤AY It is known…

Q: Let x = (13, −20, 15), v1 = (1, −2, 3), and v2 = (−1, 1, 1). Then the set {v1, v2} is orthogonal. It…

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Q: Find the norm of f(x) = √√1-x² on the interval [- 1,1]. 2√3 3 O O O W|N H

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Q: What is the dimension of the following linear space? 2 x 3 The space R²X

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Q: Given points p = and p3 = in R?, 2 P2 = let S = conv {p],P2» P3}. For each linear functional f, find…

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Q: Let (X,d) be a metric space , x ϵ X and A ⊑ X be a nonempy set. Prove that d (x ,A) = 0 if and only…

A: Given: X,d is a metric space, x∈X and A⊂ X is a non-empty set. To Prove: dx,A=0 if and only if every…

Q: a) Find a Lagrange interpolating polynomial in the form of P, (x) = ax³ + bx² + cx+d befitting the…

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Q: Let A = {AC R| A is countable} and A2 = {A C RI A° is countable} which is a measurable space? (IR,…

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Q: Extend the linearly independent Set S={x+1, 2X²} to the space base P₂ (R)

A: We are given a linearly independent set S= {x+1, 2x2} We have to extend S to basis of P2(R).    Let…

Q: (b) Let X = {0, 1} and let (X*,d) be metric space, where d(x, y) = the number of elements of the set…

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Q: Suppose A is a closed, non-empty subset of a metric space (X, d) and p a point of X which is not in…

A: Given:  A is a closed and non-empty set of a metric space (X,d) and p∈X , p∉A. To Prove: inf{d(p,a)…

Q: .. .): P, is a polynomial space with degree with order n and less than equal n. Is the space S =…

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Q: (1 x x²). Let F be the 2: Consider the polynomial space P₂ with the standard base V = corresponding…

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Q: [1030] Let A = 0 120 %3D V be the null space of A, and W= V.What is the dimension of W? 0 0 00 1 3.…

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Q: Determine the dimensions of the following spaces:    a) span ((1, 2, 3), (2, 4, 6), (3, 6, 9))…

A: In this question, we will find the dimensions of the spaces.

Q: ) Give definition to the convex set of the space Rn. Also give a definition for when the function f…

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Q: We note that ℤ2 which means ℤ*ℤ. Let d: ℤ2 *ℤ2→ℝ≥0 be the function that is given by…

A: The given in this question is a function d that takes two points (x0,y0) and (x1,y1) in the set of…