3.3.9. Given a normed space X, prove that the function p: X R defined by p(x) = ||x|| is continuous on X.

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: (2.3.15. Let X be a metric space. (a) Prove that SCX is bounded if and only if diam(S) < 0o.

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

Q: 8 Examine holomorphism for given function f(z) = |z|

A: The given problem is to examine the holomorphism of the given complex function. Given function…

Q: c) For x e ll n2, define || x ]| = || x || 1+ || x ||2: Show that (11 n2, ||· || is a Banach space.

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Q: 4. Assume that a continuous function f : R –→ R is T-periodic; that is, f(x + T) = f(x) for all x E…

A: given a continuous function f:ℝ→ℝ is T-periodic, defined as fx+T=fx ∀x∈ℝ to prove- for every natural…

Q: Theorem 6.2.3 Let X,→X and g be a continuous function on P le bre g(X„) → g(X)

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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: 1. Prove that if f(x) is bounded and measurable on E, and c is any constant, OMNIA then S, [f(x) +…

A: Given, f(x) is bounded and measurable on ETo prove: ∫Ef(x)+c=∫Ef(x) dx+c m(E)

Q: 2. Show that if f : R → R is continuous on R and f(r) = 0 for all r E Q then f(r) = 0 for all x E R.

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Q: 3. Use the e, d definition to show that f(x) = x² is uniformly continuous on A = [0,3]. %3D

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Q: Theorem 8.1. The following are equivalent: (1) X is connected. (2) There is no continuous function f…

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Q: (2) Let m(E) 1}) = 0. (ii) Show that lim, S f" = m({x € E : f(x) = 1}). E

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Q: 4:x→X (when X is a normed Space) is continous at xo € X iff for any Sequeue ie AS n s

A: Let f:x→X (when X is a normed space) is continuous at x0 ∈X iff for any sequence  xn ∈X, xn→х0  ⇒…

Q: Suppose that f is a continuous function satisfying stæ) = a f(t)dt + * So f(t)dt + a %3D for all a,…

A: Solve the function

Q: 6. Consider S = [a, b] with p(x, y) = |x – y|. Verify that T : S – S with T(x) = - and a > 1 is a…

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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: 6. Let f: A R be a continuous function. Show that if A is compact then sup f(A)e f(A).

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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: . If f is a continuous function from the metric space X to the metric space Y and {xn} is a sequence…

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Q: Check the validity of Rolle’s theorem for the functionf(x)=√1−x2on the segment [−1,1].

A: Under square root we can not have negative number So the function f(x) exists in the interval…

Q: 2. Consider the function f(z) = ( -3) on (0, 1]. (a) Prove that / for r, ye 0, 1], (b) Prove that /…

A: Contraction mapping theorem: Let (M, d) be a metric space. A mapping f:X→X is contraction mapping,…

Q: 8. C'(0, 1] is the space of continuous functions f : [0, 1] → R with a continuous derivative f' :…

A: Let, fnn=1∞ be a Cauchy sequence in C10,1than for any ε&gt;0 there exist N∈ℕ such that ∀m,n&gt;N…

Q: 5. Determine the vclue of c such that the function is continuous on (-o0,00 + flx) = 2x, x SC

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Q: Prove that the function f(x1,x2) = In (e*i+= + eV=i+1) is convex over R².

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Q: 2. Let g : R → R be a bounded function and define f : R\ {0} → R by 1 f(x) = g(x)+ x2 Using only the…

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Q: Let E have measure zero. Show that if ƒ is a bounded function on E, then f is measurable and ſe ƒ =…

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Q: Suppose x e R and let A = 1 1 x 1 x x x x x. (a) Compute det A as a function of x. How does the rank…

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Q: s(2) = f(z) = Im z is continuous ylgn O İhi O

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Q: 8 Examine holomorphism of given function  f(z) = |z|2

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Q: 4. Let X = C[0, 1] be the space of continuous real valued functions on [0, 1]. Find a subset K of X…

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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: Which of the following is the statement of Fermat's Theorem? a. If f has a local minimum or maximum…

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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: 1. Determine whether k(x) is continuous at a = 3, where x + 3, x 2.

A: Left-hand limit of f(x) at x=a is represented by limx→a- f(x) and it can also written as limh→0+…

Q: Prove: If S is compact, and f is continuous on S, then f takes a minimum value some- where in S.

A: Since S is compact and f is continuous, f[S] is a compact subset of R and therefore closed and…

Q: Suppose f,9 : D R are continuous on D. uniformly continuous on D. Prove that f +g is uniformly

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Q: 8. If f is conformal on a domain D which is symmetric with respect to the real axis, is f(z)…

A: Given that f is conformal on a domain D which is symmetric with respect to the real axis.

Q: Prove the theorem stating that a E R, if x belongs to the neighborhood V¿(a) for every ɛ > 0, then x…

A: To prove the theorem stating that a∈ℝ, if x belongs to the neighborhood Vεa for every ε&gt;0, then…

Q: 1- Suppose that ,p) is a meric space, an that E CX. Then the interior E is open (m X) and the…

A: The given question is related to metric spaces. The solution is explained below.

Q: s continuous in and f'(x) exist 11

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Q: 1 Are the following sets of functions, each defined on x E R, linearly independent? Use Wronskian.…

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Q: If ƒ : (0,1] → R and g : [1,2) → R are uniformly continuous on their domains, and S (x) for æ € (0,…

A: The given function is uniformly continuous in the intervals 0,1 and 1,0. The interval 0,2 is just a…

Q: 2.9.11 Prove that every function f: Z→ F is continuous, where the metric on Z is the restriction of…

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Q: Given, we have a finite purely non-atomic measure space (X, F, µ), prove that µ(F) = [0 µ(X)].

A: Given: Here, the given is finite purely non-atomic measure space (X,F,μ). To prove that μ(F)=0, μ(X)

Q: Find the Wronskian for the set of functions.{x, x2, ex, e−x}

A: Given information: x, x2, ex, e-x The Wronskian is given by the following determinant.…

Q: 2. Do as indicated. x +1 if x m,

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Q: 6. If af/ðæ is continuous on the rectangle R = {(t, x) : 0 0 such that |f(t, x1) – f(t, x2)| < K\x1…

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

Q: Find the Wronskians of the given sets of functions and determine whether the functions are linearly…

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Q: Theorem 7-6. If the variables are uniformly bounded then the condition, lim Bn = 0, n- 00 n2 is…

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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.