1. Convex Sets 5 Remark: See Section II.11 and, especially, Problem 3 of Section II.11.3. Often, we consider convex sets in a more general setting. (1.4) Convex sets in vector spaces. We recall that a set V with the operations “+” (addition): V ×V −→ V and “·” (scalar multiplication): R×V −→ V is called a (real) vector space provided the following eight axioms are satisfied: (1) u + v = v + u for any two u, v ∈ V (2) u + (v + w) = (u + v) + w for any three u, v, w ∈ V (3) (αβ)v = α(βv) for any v ∈ V and any α, β ∈ R (4) 1v = v for any v ∈ V (5) (α + β)v = αv + βv for any v ∈ V and any α, β ∈ R (6) α(v + u) = αv + αu for any α ∈ R and any u, v ∈ V (7) there exists a zero vector 0 ∈ V such that v + 0 = v for each v ∈ V and (8) for each v ∈ V there exists a vector − v ∈ V such that v + (−v) = 0. We often say “points” instead of “vectors”, especially when we have no par- ticular reason to consider 0 (which we often denote just by 0) to be significantly different from any other point (vector) in V . A set A ⊂ V is called convex, provided for all x, y ∈ A the interval [x, y] = αx + (1 − α)y : 0 ≤ α ≤ 1 is contained in A. Again, we agree that the empty set is convex. A convex combi- nation of a finite set of points in V and a convex hull conv(A) of a set A ⊂ V are defined just as in the case of Euclidean space. PROBLEM. 1◦. Let V be the space of all continuous real-valued functions f : [0, 1] −→ R. Prove that the sets B = f ∈ V : |f(τ)| ≤ 1 for all τ ∈ [0, 1] and K = f ∈ V : f(τ) ≤ 0 for all τ ∈ [0, 1] are convex. (1.5) Operations with convex sets. Let V be a vector space and let A, B ⊂ V be (convex) sets. The Minkowski sum A + B is a subset in V defined by A + B = x + y : x ∈ A, y ∈ B .

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