3. In each of the following examples, determine (with careful arguments) the bound- ary of A in X, and state whether A is open in X, closed in X, both open and closed in X, or neither open nor closed in X. a) x = R, A = (-1,1).

Q: Let z= y- 4a, a u'v and y-v- 3u. Compute On kum)-(1,1) -

A:

Q: 8.0. If x-(x,, X, ...,%)«R', define [xL by Prove that xJkL is a norm on R'. N I|} dns - x]

A: We will solve the first question highlighted in yellow. If you want the second question to be…

Q: 1. Prove that for every non-gero normed space X there is a non-zero bounded linear fmetional acting…

A:

Q: let f: [o,0o] - [o,00) be decined by Fla) =x? IF is a homeomor phism Detemine

A:

Q: Prove that det(kA) = kn det(A) for any A ϵ Mnxn(F ).

A:

Q: Prove that the map g : Z →→ N defined by | 2k, -(2k+1), k < 0 k 2 0 f (t) = is one to-one nd mans…

A:

Q: a) Assume that y is a continously functian in t. Evaluate differ entiable TT to

A: Given ∫xexπyty'tdt

Q: 2.4 (Ross, 2003, page 593) Show that the relationship $ (x) = x;¢(1;, x) + (1 – x;)ø(0¡, x) is true…

A:

Q: Let n, k, m EN with n2k21. prove that

A:

Q: - Show that the norm tion t= aT+B. Use [a, b] onto [0, 1] (7) = 1, ý(7)= T, V т,

A:

Q: 1. Prove by Induction that the numbr f edges of Kn

A:

Q: Compute the orthaganal piojectian of ? onto the lina L through -6 and the origin

A:

Q: Prove that the set S = {n² : n e N} is denumerable. %3D

A: We have to show that the given set S is a denumerable set.

Q: 1. Identify which theorem will prove that ZEDC = ZDEC= ZECD if DE EC = CD. A,

A: Given problem:-

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 T2 is chaotic on 1.

A: Given that T2=2x0&lt;x&lt;122(1-x)12&lt;x&lt;1 The maximum for T is 1 at x=0.5. The dynamics of T2…

Q: Let po) - X In (x) , using the following approx im for the secoud denvative fcxo) = FOXO -h)-ZR…

A: Introduction: Apart from differentiating a function analytically, the derivative of a function can…

Q: Exercise 6.40. Let E be a nonempty subset of R that is bounded above. Prove that if a = sup E, then…

A: This is the question of topology.

Q: 9. Let be a bounded open subset of C, and y: → a holomorphic function. Prove that if there exists a…

A:

Q: 3.2 Consider R2 with (, ·) defined for all x and y in R? as х and 2 (x, y) := x |1 =:A Is (:, :) an…

A: The objective of the question check for inner product.

Q: 10. If A is 3 x 3 with rank A = 2, show that the dimension of the null space of A is l.

A:

Q: If x is a lower bound for S in R then x must be unique..

A: If x is a lower bound for S in R then x must be unique.

Q: @ Let A -a Find Cenditiens on a,b.c and d that AX =o has infailely many Solution. what must there…

A:

Q: 9. Let be a bounded open subset of C, and 4: Prove that if there exists a point zo E such that 2 a…

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: Show that if f and g are bounded and satisfy a Lipschitz condi- tion on an interval, then f g…

A:

Q: 32. Show that, in a normed space, x x implies -(x1 ++Xn) → x.

A:

Q: c If in is an involution of X, show that (x, y)in = (yin, xin) is an involution of X2). %3D

A: Let X be a lattice We have to show that (x, y)in=(yin, xin) is an X[2] X[2] is a 2 x 2 matrix We…

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: 2. Let f : X → Y be onto. For each b e Y, let A, {As : b € Y} is a partition of X. f6]. Prove that

A: Given that f:X→Y is an onto function. Then for every b∈Y there exists x∈X such that  fx=y Also,…

Q: (3) In R² with the standard topology: {(x, y) | x² + y² {(x, y) | x² + y? > 1}. = 1}, {(x, y) | x² +…

A: In a topological space R2with standard topology. A  subset O is open if at each point x in O there…

Q: 1. Let T and S be topologies in X. Show that SnT is also a topology in X.

A: 1.

Q: 8. (a) Suppose f : [a, b] → R is integrable and L(ƒ, P) = U(ƒ, P) for some partition P of [a, b].…

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: let Chn),(t) be dearuences of bounded Inal lenvegas uni formly repectively Show functrons. on A Lo…

A: Given - Let hn, tn be sequences of bounded functions on A that converges uniformly on A to h, t…

Q: 1. Prove that any quasigroup satisfying one of the identities {(ry)x satisfy the other. = y, z(yr)…

A:

Q: Suppose U and v are conformally equivalent. Prove that if U is simply connected, then so is V.…

A: Given that U and V are conformally equivalent that is there exist a mapping f from U to V such that…

Q: of a space X and let x e X. Then x e A iff there exists a net in A (that is a net whi to x in X.

A:

Q: Theorem 6.17. Every compact subset C of R contains a maximum in the set C, i.e., there is an m E C…

A: To prove that, every compact subset C of ℝ contains a maximum in the set C, that is, there is an m∈C…

Q: 3. DETERMINE THE UOLUME OF THE REGION BOUN DED ABOUE BY THE PARABOLOID Z=2-x²-y BY THE CON E

A:

Q: prove that Sa/Gn has f discription with M-ly ny-l Degrees tof freedom

A: Suppose that xi and yi are two random samples of size n, drawn from a normal population.  Population…

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: 8. Show that A is Hermitian if and only if A = B - C with B,C positive.

A: A is Hermitian if and only if A=B-C with B, C positive.

Q: 11 Prove that if A CR and |A| > 0, then there exists a subset of A that is not Lebesgue measurable.

A: As per the policy, we are solving a first question. Please repost the question and specify which…

Q: 4.4 Prove that if A is nowhere dense in discrete space (S, d) then A = 0.

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

Q: let FC k be a Galois extension with Gal

A:

Q: Let X = (!,2) and T3} U = (1,2-)inel,2,U+ jult. %3D n+1 show that X is not T - space

A: This is a problem of Topology.

Q: 1/ For any. topological space (X, T), if T has only three elements then (X, T) is connected.

A: Given: Topological space X, T and T has 3 elements  To prove that X, T is connected when T has 3…

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: If e-ispure imaginary W hat Restriction is placed on z?

A:

Q: Let A = {x E Z* : x is a mutiple of 2 and x < 15}. Rewrite A in roster notatlon.

A: Given that : A = {x∈Z+ : x is a multiple of 2 and x &lt;15 }   Roster form : A representation of a…

Q: 4.2 Let Fj, be a closed set for k = 1,2, ... ,n in (S, d). Show that U Fk is closed.

A: