Show that if f and g are bounded and satisfy a Lipschitz condi- tion on an interval, then f g satisfies a Lipschitz condition. Give a counterexample to show that it is necessary to assume bound- edness.

Q: Example 14. If f (x) = 0, for every x in the Cantor set P and f (x) = k for x in each of the…

A:

Q: show that a bounded Lineart on a complex Hilbert spase H is normal (i-eTT* = T*T) if and only if 11+…

A: We have to show that T  is normal if and only if ||T* x || = || Tx|| for all x € H. Using this, to…

Q: Assume that A and B are nonempty, bounded above, and satisfy B CACR. Show that sup B < sup A.

A: Introduction: The supremum and infimum properties are the very important properties of any subset of…

Q: Qu:- Prove that funcdiod U = ay - 2x -2 xy-j+x* is hav monic . Firel - funchion v such thut fcas…

A: Solution

Q: Define f : Zmn → Zm × Zn by ƒ ([x]mn) = ([x]m, [x]n). Show that f is a function and that f is onto…

A: The function f:ℤmn→ℤm×ℤn is defined by                  fxmn=xm,xn Show f is a function and f is…

Q: Еx6. IS V is finite dimensional ond f is a homomor- phisni of V into itself which is not onto prove…

A: If f is a homomorphism of V into itself, then f(0) = 0 .

Q: Let a ≠ b in a field F. Show that x + a and x + b are relatively prime in F[x].

A: Definition of relatively prime: A polynomial in fx and gx in Fx is said to relatively prime if the…

Q: Prove (sn) is bounded if and only if lim sup |Sn| < +∞.

A:

Q: 2. Show that A is Hermitian if and only if (|A|) ER for all |t).

A: The given question is related with functional analysis. We have to show that…

Q: Provethat amappi'ng from a metric SpaceXto metri'c space X is con tinuok ifand only f the inverse…

A:

Q: Find a counterexample to Lie's Theorem when k is not algebraically closed and chark = 0. %3D

A: According to Lie's theorem, for any finite-dimensional linear representation of a finite-dimensional…

Q: Let T₁, T₂, be the following bounded linear operators ¹ T₁(x₁, x2. X3,) = (x₁, X₁, X₁, X₁, ...)…

A: Here, in the question, there are given the bounded linear operators l1→l∞ which are given as…

Q: The set of integers is not dense in cofinite topo log y of R True False O

A: The set of integers is not dense in co finite topology on ℝ.

Q: Consider the set A = {1/2^{n-1}: n\epsilon \mathbb{N}}. Show that Inf(A) = 0. WITHOUT USING LIMITS

A:

Q: Show that a linear map T:X-→ Y between normed spaces is continuous if and only if {Tx,} is a Cauchy…

A:

Q: (a) Prove that the norm limit of a sequence of compact operators is compact.

A: Solution :-

Q: Suppose both f(z) and the conjugate of f(z) are holomorphic on a connected open set.  Show that f is…

A:

Q: 2. Let ACR. If a is a lower bound of A, then it follows that -a is an upper bound for A.

A:

Q: Let A ϵ Mnxn(F). Under what conditions is det( - A) = det(A)?

A: To Determine: Let A ϵ Mnxn(F). Under what conditions is det( - A) = det(A)? Given: we have a matrix…

Q: prove that an is bounded above.

A: In this question, we prove the sequence  an=(1+1n)n is convergent by given porcedure. i.e.…

Q: Prove C(x) with the supremum norm is a metric space by showing the supremum norm is a metric.…

A: Uniform convergence implies the pointwise convergence of the sequence.

Q: Let f,g : R → R be bounded and uniformly continuous. Prove that fg is also bounded and iniformly…

A:

Q: Let X be a set, let fn : X → R be a sequence of functions, and let f : X → Y be a function. Let ē…

A:

Q: Show that the Modulus Function            f : R → R, given by f(x) = | x |, is neither one-one nor…

A: Consider, f: R → R given by f(x) = | x |Where,| x | is x, if x is positive or 0| x | is – x, if x is…

Q: Show that (6.g) = (0,0) is unijomy continuous or not.

A: Uniform continuity

Q: :- Prove that funcdion U = 3y - 2x -2xy-j+x is har monic . Final • funchion v such thut is andilytic…

A: Given function: u=x2-y2-2xy-2x+3y

Q: 1. Suppose that f(z) is analytic on 0 < |z| < ∞ and there exists a f(z) nonnegative integer m such…

A:

Q: Prove that det(At) = det(A) for any A ϵ M2x2(F).

A: Suppose we are given a square matrix A with four-element: a,b,c,d A=abcdAt=acbd

Q: Let K be a compact subset of a metric space X, and assume that -R is continuous. Prove that f…

A:

Q: 6) let fl2) = CosCitaiy) 22atey Express flz) that f satisfies in the form and fornd z So the Canchy-…

A:

Q: IS V is finite dimensiona! oad f is a homomor- Ex 6. phisni of V into itself which is not onto prove…

A: Given: V is finite dimensional and f is homomorphism of V into itself such that f is not onto. To…

Q: Let V and W be two Banach spaces and (Tn) a sequence in the space of bounded linear maps from V to…

A:

Q: If an upper bound exists, then it is unique.  true or false

A: An upper boond of a set is an element which is larger or equal to all elements of set A. For…

Q: Prove that if a is an upper bound for A, and if a is also anelement of A, then it must be that a =…

A: We have to prove that if a is an upper bound for A, and if a is also an element of A, then it must…

Q: If f and g are uniformly continuous and bounded, show that f.g is uniformly continuous. Give a…

A:

Q: . Let X be a r.V. such that E(X") is finite for all n = 1, 2,... Use the expansion

A: Introduction: Suppose X is a random variable. Then, the moment generating function (mgf) of X is…

Q: Show that if a contraction map f: from M to M has a fixed point then it's unique.

A: Definition: Let X, d be a metric space. The map T: X→X is said to be a contraction mapping if there…

Q: If (an) has limit−1 and (bn) tends to infinity, does it follow that only finitely many terms of (an…

A: No it doesn't follow the statement

Q: Let a > 1 be an integer. Prove that n|p(a" – 1), where o is the Euler ç-function.

A: In number theory, Euler's totient function counts the positive integers up to a given integer n that…

Q: ntin

A: It is given that,  If g is continuous and ∫-28g(x)dx=30 We have to find ∫-144g(2x)dx=?

Q: Suppose f : (M, d) → (N, p) is continuous. Prove or disprove: f is uniformly continuous + f maps a…

A: f: (M,d)→(N,ρ) Proof: fis uniformly continuous ⇔ f maps a cauchy seq. to a cauchy seq.

Q: Prove that f(n) = 1000n5 + 20000n2 + 32 is O(n6 ); Is this a tight upper bound? Why or why not?

A:

Q: Suppose functions w: PQ and h : Q+ R are both onto. Prove that the composite function is also onto.

A:

Q: 1. If f:X → Y and g:Y → Z are functions and gof is onto. a. Must g be onto? Give a proof or a…

A: (a) Given that  f:X→Yg:Y→Z and g∘f is onto To prove : g is onto Let x∈X, y∈Y , z∈Z As gof is onto…

Q: Given a closed set A, construct a continuously differentiable func- tion that has A as its set of…

A:

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

A:

Q: Prove that det(kA) = kn det(A) for any A ∈ Mn×n(F).

A: In determinant the scalar quantity is taken common from a single row or a column.