My mentor has assigned me the task of studying the content of Appendix A and part of Appendix B in Bertseka's Nonlinear Programming, which covers the basics of convex analysis and its prerequisites. The appendage cycles through closed and open sets, eigen and square arrays, symmetric and positive definite matrices, iterations through the mean value set, the second order extension, the descent model, the implicit function set, the contraction mapping set, and the set of functions. Caratheodory's theorem, several features of closing and continuity up to the projection set. But I do not know most of it, and I still have a significant part to cover, and that's why I want to help with material and practice.

The resources that I like most would preferably have many examples and exercises, they could cover content that would not help me with the attachment, but in that case you could determine which sections are worth reading. The same is not true for them All the topics that I have mentioned, but knowing what is not being treated is helpful. Ideally, I like lyrics that provide reflections that sharpen your intuition, change your mindset, help with visualization, or simply add examples and empower your practice.

To understand what I want is an example that has all of the above aspects:

In Spivak's book, Calculus on Manifolds, in which he introduces the definitions of inside, outside, and boundary, he points out that these contrapuntal meanings have what you will understand in the exercises he mentions.

If $ A subset Rn $ and $ x in R ^ n $, then one of three options

must hold (Figure 1-2):

- There is an open rectangle $ B $ so that $ x in B subset A $,
- There is an open rectangle $ B $ so that $ x in B subset R ^ n – A $,
- If $ B $ is any open rectangle with $ x in B $then B contains points from both $ A $ and $ R ^ n – A $,

The points that satisfy (1) form the **Inner** from $ A $, those

satisfactory (2) the **Outside** from $ A $and those who (3) the

**border** from $ A $, Issues 1-16 to 1-18 show that these terms may occur

sometimes have unexpected meanings.

Recommended exercises:

**1-16.** Find the interior, exterior and boundaries of the sets:

**1-17.** Set a set $ A subset [0,1]times[0,1]$ so that $ A $ contains at most one point on each horizontal and vertical line

but limit $ A = [0,1]times[0,1]$, *Note:* It is enough to ensure

The $ A $ contains points in every quarter of the place

$[0,1]times[0,1]$ and also in every sixteenth etc.

**1-18.** If $ A subset [0,1]$ is the union of open intervals $ (a_i, b_i) $ so every rational number in (0.1) $ is included

something $ (a_i, b_i) $Show this limit $ A = [0,1] – A $,

After that, I got a clearer picture of what those three terms mean. I had a similar experience with Cauculus of the same author and with Linear Algebra by Hoffman and Kunze, who went through the concepts that show how linear systems, matrices, and linear transformations have a complicated equivalent.

An additional feature missing in the appendix is the connection between the various topics.

I am sorry for the really long and possibly unclear question. I hope I did not give the impression that I'm strict, which is a good suggestion. If you think your proposal does not fit into some categories, just publish it. It could be anything, an exercise book, a book, a short text, anything.