Let U be a family of real-valued L-Lipschitz functions on some metric space X. Suppose that there is some real number b and some xo EX such that b ≤ u(xo) for all u EU. Define U: X→ [-∞, +∞] by U(y) = inf{u(y) : u € U}. Show that a) U(X) CR. b) U is L-Lipschitz.

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A: It is given that f is holomorphic in the disc with centre 0 and radius 2. So, the function f is…

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Q: Q7: Prove that (R", |l-|| ) is complete metric space.

A: This is a problem of Topology, Functional Analysis.

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A: I have proved the three conditions for metric

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A: A function f is said to bijective if it is one-one and onto

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A: Given that, P denotes the set of functions f in C0,1

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A: This is a problem from topology.

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

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A: The given question is epimorphism. Given that,      f : M2ℤ , + , · → M2ℤ , + , · is a function…