By Andrew J. Kurdila, Michael Zabarankin

This quantity is devoted to the basics of convex practical research. It offers these features of useful research which are generally utilized in a number of functions to mechanics and keep watch over thought. the aim of the textual content is largely two-fold. at the one hand, a naked minimal of the speculation required to appreciate the rules of practical, convex and set-valued research is gifted. a variety of examples and diagrams supply as intuitive a proof of the foundations as attainable. nonetheless, the quantity is essentially self-contained. people with a heritage in graduate arithmetic will discover a concise precis of all major definitions and theorems.

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**Example text**

2 Thus, x ∈ B(xk , rk ) ∀k and x∈ / Nk ∀ k. Consequently, we conclude that ∞ x∈ / Nk = X. k=1 This is a contradiction and the theorem is proved. 3. 5 Compactness of Sets in Metric Spaces The next topic in our review of metric spaces is a discussion of compact sets. As we will see later, compactness plays a crucial role in establishing uniform continuity in many applications. 12. A subset A contained in a metric space X is sequentially compact if any sequence extracted from A {xk }k∈N ⊆ A contains a subsequence that converges to some x0 ∈ A.

3. Suppose (X, τx ) and (Y, τy ) are topological spaces. The mapping f : X → Y is continuous at x0 ∈ X if and only if the inverse image of every open set Qy ⊆ Y with f (x0 ) ∈ Qy contains an open set Qx ⊆ X such that x0 ∈ Qx ⊆ f −1 (Qy ). Proof. Necessity. Let f be continuous mapping X → Y at x0 and let Qy be an open set, containing point y0 = f (x0 ). But it means that Qy is a neighborhood of y0 . 15, there exists a neighborhood Qx ∈ X of point x0 such that f (Qx ) ⊂ Qy , that is, x0 ∈ Qx ⊂ f −1 (Qy ) and Qx is open.

A compact subset of a Hausdorﬀ space is necessarily closed, so we have K = K. It follows that A⊆K =⇒ A⊆K =⇒ A ⊆ K. Since in a Hausdorﬀ space a closed subset of a compact set is compact, we conclude that A is compact and, consequently A is relatively compact. The next theorem characterizes relatively compact sets in metric spaces. 4 (Hausdorﬀ ). A set A is relatively compact in a complete metric space X if A is totally bounded. We see that a set A is relatively compact in a metric space X if every sequence in A contains a subsequence that converges in X.