a) Show that every coherent risk measure o: MR is a convex risk measure, that is g satisfies o(AL₁ + (1-X) L₂) ≤ Ag(L₁)+(1-X)g(L2), [0, 1], L₁, L₂ € M. b) Show that for positive-homogeneous risk measures convexity and coherence are equivalent. c) Show by means of a counterexample that without positive homogeneity the statement in b) is false.

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A: Using simplex pivot tableau method

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A: The solution is given below in the next step:

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A: Answer:----.                 Date:----8/10/2021

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Q: 1. For T € L(V,W), prove that null(T) = (range(T*)) and range(T) = (null(T*))+.

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