2.9.11 Prove that every function f: Z→ F is continuous, where the metric on Z is the restriction of the standard metric from R to Z.

Q: Let X be a normed space and aɛ X. Show that there exists an fe X' such that ||a|| = Sup { \f(a)| :…

A: Given that,                X is a normed space and a ∈ X . We have to show that there exists an f ∈…

Q: Let X = [0, 1] U [2, 3] endowed with the standard metric. Then [0, 1] is an open subset of X. Select…

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Q: 4. Suppose that (X, dx) and (Y, dy) are metric spaces, and define distance on X × Y by Prove that…

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

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: Suppose f(z) is analytic in the unit disk D={|z| < 1} and |f(z)|<= 1 for |z| <1, Prove that…

A: Assume that f(z) is analytic on the unit disk D ={z: |z| < 1}.Also assume that |f(z)|≤1 for…

Q: where a, b e X, is also a metric on X. 2- Let (X, d) be a metric space and T the corresponding…

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Q: 5. Let d be a metric on X. Determine all constants k such that (i) kd, (ii) d+ k is a metric on X.

A: Given that d be a metric on X. Then d:X×X→[0,∞) satisfies the following conditions for x,y,z∈X, (a)…

Q: Consider X = C[0, 1] with the norm = max |x(1)|, IE[0,1] and let T: X - X be the mapping given by…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: 16. The set S = { x∈R: x2 - 4<0} with the usual metric is

A: The set S = { x∈R: x2 - 4&lt;0} with the usual metric is ..

Q: 4. i) Let X = B [a, b] and let d be the sup metric on X. If n → x in (X,d) and an is continuous at…

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Q: = 0,1] and d be the usual metric, d(x, y) = a -y %3D en d can not induces a norm on X True False

A: the usual metric d(x,y)=|x-y| usually induces the norm.

Q: b) Let X be an inner product space and x, y e x.Prove that xly if and only if kx + y = |lk | + |y*k…

A: Given that X be an inner product space and x,y∈X Let, x⊥y. The objective is to prove that…

Q: E be a Euclidean space of dimension (E(1, f)) = dim(Ker (f- id)), then s minimal.

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Q: Suppose Show that p: X²> t0,00) defined by d is a metric n X, 2. 3d (x,y) 2+3d (xy) metric an p…

A: Given that d is a metric on X. We need to prove that ρ is a metric.

Q: Example (3): Let (M, d) be a metric space. Define a function e: Mx M→R by: e(x,y) = Min{1,d(x y)};…

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Q: (b) Let (M, d) be a metric space. Define a function e: MXM-R by e(x, y) = min {1,d(x, y)}; vx,y E M.…

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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: Exercise 2. Suppose V and W are finite-dimensional, T E L(V, W), and there exists PE W' such that…

A: Given V and W are finite dimensional , T∈LV,W and there exists φ∈W' such that nullT'=spanφ We have…

Q: Show that X and Y are independent iff E(f o X – go Y)² E(f o X – E(f o X))² for all Borel-measurable…

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Q: Given any Hermitian space E with › E, if (f(x), x) = 0 for all x € E, th I

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Q: 3.4 Give the definition of a separable metric space (S, d). 3.5 Show that the metric space (R, d),…

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Q: (b) Consider X = C[0, 1] with the norm %3D = max |x(1)), re[0,1] and let T: X X be the mapping given…

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Q: Question 4. Let f:X Y be a mapping from a topological space (X,7x) a topological space (Y,Ty). Prove…

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Q: 4. Let (x, d) be a metric space and ACX be nonempty set. Define f:X → IR, fix)= inf dex,a). aE A…

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Q: i) Pe(x) = 0 iff r € Ë. ii) Prove that pe is uniformly continuous.

A: i Let E is the non-empty subset of a metric space X,d the distance from x to E is defined as…

Q: Let X = R, and d be the discrete metric, and A = [3, 6) Show that if A is compact or not

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Q: 3. i) Let (X, d) be a metric space. If æn → x in (X, d) and y e X, prove that lim d(xn, y) = d (x,…

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Q: 1: Let (X,) be a metric space and let d1(x.,y)=K d(x.y), K>0. Prove that d1 is a metric function on…

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Q: E be a subset of a normed space X and Y = Span E and a € X. Show that Let a € Ỹ if and only if f(a)…

A: To find: a∈Y¯, and f(a) = 0 whenever f∈Xι and f is 0 everywhere on E.

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: For every ,x, y E R, let p(x, y) =| x² – y² |. Is p a metric -

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Q: Given the discrete metric space for n E Na the discrete metric o defined by the distance function 0…

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Q: 3.27.2. Let X be a metric space with metric d; let ACX be nonempty. (a) Show that d(x, A) = 0 iff x…

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Q: Given the family C[0, 1] of continuous functions ƒ [0, 1] → R, and the norm ||f|| = sup |f(x)]. x=…

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Q: Question 4: Suppose that the metric d on M is the discrete metric. (a) Show that every map f: M →Y…

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Q: Let X and Y be metric spaces. Assume that E is a dense subset of X, and f, g: X Y are continuons…

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Q: 9) Suppose V is a vector space and S,T E L(V,V) are such that Range S c ker T =null T. Prove that…

A: See the detailed solution below.

Q: 2.3.11. Let d be the discrete metric on a set X. Explicitly describe the open balls B,(r) for all a…

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Q: 7. Let (S, p) be a metric space and T : S → S be a contraction mapping. Prove that T is uniformly…

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Q: Let X be a compact subset of R and let C(X) denote the set of continuous real-valued functions on X…

A: let X be a compact subset of ℝ and let CX denote the set of continuous real valued functions on X.…

Q: Please solve this linear algebra question. Thank you.

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Q: S={xER:x2 - 4<0} with the usual metric is A. Compact. B. connected. C. Not connected D. Sequentially…

A: Given set  S= {x€R : x2 -4&lt;0}

Q: (b) Given two metric spaces X,p>, and a function f : X→ Y that is uniformly continuous on SC X. If…

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Q: Given the family C[0, 1] of continuous functions ƒ : [0, 1] → R, and the norm ||f|| = sup f(x)]. x=…

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Q: Let A be a subset of R with the usual metric such that A=A. If there exists r>0 such that d(x,y) <r…

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Q: A mapping f:C→ C is called bounded, if sup |f(2)| < o0. zEC Show that sin, cos : C →C are unbounded…

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Q: Let f: (N,d) (N, d), where d is the usual metric and di is the discrete metric, defined f as f (x)…

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Addition Rule of Probability

It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.

Expected Value

When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).

Probability Distributions

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

Basic Probability

The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.