Show that a linear map T:X-→ Y between normed spaces is continuous if and only if {Tx,} is a Cauchy sequence in y for every Cauchy sequence {x, }in X .

Q: any Hermitian positive definite matria A=M-N, + N is Hermition, and if it is positive

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Q: For X and Y normed linear spaces, let {n} be a sequence in L(X, Y) such that T →T in L(X, Y) and let…

A: Please find the answer in next step

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…

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Q: Let (h,), (tn) be sequences of bounded functions on A that converges uniformly on A to h, t…

A: Given that there are two sequence (hn) and (tn) be sequence of bounded function on A that converges…

Q: Let X be a Banach space an {xn} be a sequence in. X. Prove that if {xn} converges in norm in X, then…

A: Given: Suppose X is a Banach space and xn→x in X. To prove: xn→x weakly

Q: Let X be a normed linear space over F. It follows from Exercise 1 that X* B(X,F) is a Banach space.…

A: Given that X is a normed linear space over F and X* = BX , F is a Banach space. ( X* is called the…

Q: Prove that the family of all polynomials of degree < N with co- efficients in (-1, 1] is uniformly…

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Q: space is the union of a finite number of closed balls radius e. Prove that a metric space is totally…

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Q: Consider f : X →Y, with domain X = {d,n, t, o, c, p, 8, j, r, a, z, g} and codomain Y = {83,11, 15,…

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Q: Give an example of a metric space X, d and a sequence {Pn} C X such that {pn} is a Cauchy sequence…

A: Let X=(0,∞) Define d:X×X→ℝ+∪0 by d(x,y)=x-y.i.e. d is a standard metric on X induced from ℝ.Take…

Q: 1. Suppose that {an} and {b.) are bounded sequences. Show that lim sup an lim supbn. lim sup an +bn…

A: As per norms, the first question is answered, To prove the property of lim sup under addition of…

Q: Let c be a cluster point of a subset A of R and let f : A → R be a function. Suppose for every…

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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: ) Let E be a non-empty compact (closed and bounded) subset of RP. Prove that every sequence X = (Tn)…

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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: given let X be a set , let fn:X→ℝ be a sequence of function and let f:X→Y be a function. let ρ¯ be…

Q: 6. Let (x,) be properly divergent and let (y,) be such that lim(x,y,) belongs to R. Show that (y,)…

A: To show that (yn) converges to 0, given (xn) is properly divergent and (yn) is such that lim(xnyn)…

Q: If fn is a uniformly equicontinuous sequence of functions on a compact interval and fn → f…

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Q: converges weakly to x in a normed linear sp

A: Given that, xn converges weekly to x in a normed linear space X We have to show that Every…

Q: 2. Prove that (1,1, 1, ...) converge to 1 in (R, ID. 3. Prove that the onlu

A: Convergence of the constant series

Q: Prove the following analogue of Theorem 3.10(b): If {Ein} is a sequence of closed nonempty and…

A: given,En is a sequence of closed nonempty and bounded sets in a complete metric space X. En⊃En+1 and…

Q: For X and Y normed linear spaces, let {T} be a sequence in L(X, Y) such that Tn → T in L(X, Y) and…

A: For X and Y normed linear space  Let { Tn}  be a sequence in L(X, Y) such that Tn → T in L(X, Y)…

Q: Show that the sequence of functions {f.} where f,(x)=: 1+ nx „xER is non-uniformly convergent on…

A: Show that the sequence of functions fn(x), where  fnx=x1+nx2,  x∈ℝ is non-uniformly convergent on…

Q: Prove that if C, D C R" are disjoint closed convex sets (but C - D is not necessarily closed), then…

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Q: Suppose {f,} is a sequence in Banach space B s.t. ||fn - fn+1|| s'/2-). Show that the sequence {f,}…

A: For a vector space X, over a field F, a normed linear space X,· is referred to as Banach space if…

Q: Question 2. space is the union of a finite number of closed balls radius e. Prove that a metric…

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Q: Show that the sequence of functions on the domain [-1, 1] defined by if -1<r<0, 0, if 0 <x < 1/n, 1,…

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Q: the sequence {fn} converges uniformly to f on R. Prove that the limit function - R be bounded.…

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Q: 1. Show that any interval (a,b) in R with the discrete metric is locaaly compact but not compact

A: note : as per our guidelines we are supposed to answer only one question. Kindly repost other…

Q: 4. Let f : DCR → R be uniformly continuous and {rn} be a Cauchy sequence in D. Then show that…

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

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Q: Let X be a Banach space and let (T,} be a sequence of bounded linear operators on X such that Tx=…

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Q: 1. Let d : R x R → R be defined to be d(x, y) = | arctan(x) – arctan(y)|- Show that d is a metric on…

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Q: 2) Prove that rxz(1) <rzz(0), where r denotes the autocorrelation function of a sequence r(n) and I…

A: Autorrelation function is mathematically given by, rxx(l) = E[X(t+l).X(t)] , for all t. Similarly,…

Q: Let w(z) be a continuous pomitive function in a bounded domain in . Define Verify is an inner…

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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: a finite dimensional normed linear space X, any subset M is compact if and only if M is closed and…

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Q: Let f : D ⊆ R → R be uniformly continuous and {xn} be a Cauchy sequence in D. Then show that {f(xn)}…

A: Uniformly Continuous If f: A  → R is uniformly continuous on A, then for given ε &gt; 0 there is δ…

Q: 2. i) State the definition of a Cauchy sequence in a metric space (X, d). When do we say that (X, d)…

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Q: Prove that a subset of a metric space is closed if and only if contains all its cluster points. Let…

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Q: 8. If A is not totally bounded, show that A has an infinite subset B that is homeo- morphic to a…

A: 8. It is given that A is not totally bounded.

Q: A topological space is compact if and only if everyultrafilter in it is convergent.

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Q: Compute the z-transform of the sequences and determine the corresponding region of convergence x(n)-…

A: For the sequence u(-n), the transform is: Z[x(n)] = X(Z) = ∑n=-∞∞x(n)z-n=∑n=-∞∞u(−n)z-n=∑n=-∞0(1)z-n…

Q: Let (an) %= (0,승,0.. be sequence in R, dis discrete metric then (xn) is

A: The sequence (xn) is neither Cauchy nor converges.

Q: Suppose {f,} is a sequence in Banach space B s.t. |[fn - fn=i|l s /2-). Show that the sequence {f,}…

A: This is a problem of Functional Analysis. Banach space.

Q: Which of the following statement is true with respect to z-transform of a right sided discrete-time…

A: The objective is to choose the correct statement about the z-transform of a right sided discrete…

Q: Consider f : X →Y, with domain X = {c,j,v, m, e, s, n, q, u, d, i, g} and codomain Y = {80,08, 57,…

A: SOLUTION-

Q: . Let ƒ : (0, 1) → R be uniformly continuous. Let (Tn) be a sequence in (0, 1). Suppose that lim, s0…

A: Given that   f:(0,1) →ℝ be uniformly continuous function. And (xn)∞n=1 be a series in (0,1) such…

Q: (8) A space X is connected space IFF every continuous funetion f:X → D, Where D-(0,1} with discrete…

A: a) Constant function