Suppose (X, d) is a compact metric space and f: X → R is continuous. Prove that f is bounded above and its supremum is attained at some point in X

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

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Q: Suppose g : [a, b] → R is continuous except at x_0 ∈ (a, b) and bounded. Prove that g ∈ R(x) on [a,…

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Q: Suppose X is a topological space, Y is a Hausdorff space, and f: X → Y is continuous. Show that the…

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Q: If a subadditive functional p on a normed space X is continuous at 0 and p(0) = 0, show that p is…

A: Given: A sub additive functional p on a normed space X is continuous at 0 and p0=0. To prove: p is…

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Q: Let V, W be finite-dimensional inner product spaces and T € L(V, W). Prove: (a) If T is injective,…

A: Injective Function:  A function is said to be injective if each element of the codomain is mapped to…

Q: Let (X, d) be a metric space. Suppose that (xn) is a sequence of points in X which converges to a…

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Q: Let (X,7) be a topological space and ACX. Show that (n) A is open if and only if Int(A) = A. (b) A…

A: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: Let f(x) be a bounded, non-negative differentiable function on [0, ∞). Prove that the length of f(x)…

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

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Q: For real numbers m and b with m + 0, show that the map f: R-→ R defined by f(x) = mx + b is…

A: Given m and b be the real number with m≠0 and f:R→R defined by fx=mx+b. We have to show that fx is…

Q: Let (X, T ) be a topological space, (M, d) be a complete metric space and BC(X, M) := {f ∈ C(X, M);…

A: Given as X,T be a topological space,M,d be a complete metric space and BCX,M=f∈CX,M;fX is bounded…

Q: Prove that if f : V → W is differentiable at a point a ∈ V it must be continuous there.

A: I have proved limiting value is equal to functional value

Q: Assume that (M, d) is a compact metric space. Show that if f: (M,d) → (Y, d) is continuous and…

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Q: If f is continuous and periodic on R, prove that f uniformly contiuous on R.

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Q: 1. Let (X, d) be a metric space and ƒ : X → X be a function. We say that ƒ is a weak contraction if…

A: Let  be a metric space and  be a function. is weak contraction if, for all..

Q: 4. Let X and Y be metric spaces and f : X → Y be continuous. Let SC X. Show: (b) If S is compact,…

A: First, we prove that fS¯⊆f(S)¯. Given that f is continuous and S¯ is compact, we can say f-1Y-f(S)¯…

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Q: g(x)]dx, (b> a), for all f, g € X. Show that (X, d) is a me

A: Matric Space

Q: State whether each of the following statements is true or false, in either case…

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Q: Prove the following theorem by defining continuity in topological space. f: (X, τ) -> (Y, V) is a…

A: The function defined on a topological space is continuous if for every open set U in Y, f-1U is open…

Q: 4. Let C₁ be the set of all continuous real-valued functions on I = [0, 1]. (a) Show that d : C₂ ×…

A: According to our guidelines I can answer only Question no. 4(a).

Q: For a metric space (X, d) and a continuous functionf from X into itself, show that the set of points…

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Q: metric

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Q: Let f be a real-valued function which is bounded on (a, 6), P be a partition of (a, b] and sup{f(r)…

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Q: Let (X, d) be a metric space, f,g:X → R some functions and xo E X, q E R. Assume that f(x) = g(x)…

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Q: Let U be a family of real-valued L-Lipschitz functions on some metric space X. Suppose that there is…

A: Let U be a family of real-valued L-Lipschitz functions on some metric space X.Suppose that there is…

Q: Let (X, d₁) and (Y, d₁) be metric spaces. Let f: X Y be continuous function, then f-¹(G) is open in…

A: Solution

Q: Let (X,dX) and (Y,dY) be metric spaces. If the sequence (fn) of functions from X to Y converges…

A: Yes, it is always true. See the proof.

Q: (h) The set of natural numbers as a subset of real numbers with the usual metric is compact. (1) Let…

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Q: Suppose that f1 : [0, 1] → R and f2 : [0, 1] → R are continuous everywhere and that f1(0) f2(1).…

A: Intermediate value theorem : Suppose be a continuous function then it takes all values between .

Q: Exerise: Letf:K⊂Rn→Rbe a continuous function. Let&be a compact set \[ \begin{array}{l} M=\{x \in x:…

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Q: Let f:R" → R be a continuous function, and let x1, x2 € R" be distinct. Prove that there exist a…

A: Given: Now consider the given, Let f:ℝn→ℝ be a continuous function, and x1, x2 ∈ℝn  be two…

Q: Prove that if g′ is bounded on (a, b) then g is uniformly continuous on (a, b).

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Q: 2. Let [0, 1] := {ƒ : [0, 1] → R: f is a bounded function on [0, 1]}. Let ||f|| ∞ : sup f(x)| x[0,1]…

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Q: very C

A: Given as, F:X→Y be linear. Cauchy sequence  Fxnis Cauchy in Y.

Q: State the principle of uniform boundedness. Use it to show that a set E in a normed space X is…

A: Sol:- The principle of uniform boundedness, also known as the Banach-Steinhaus theorem, is a…