# Real Analysis/Manual of Style

| ←Foreword | Real Analysis Manual of Style |

## Introduction

Welcome to the Real Analysis Wikibook, where we provide meaning behind all of the earlier mathematics that came before and prepare you with the eyes and sense to ready yourself for higher mathematics.
This page will discuss the feel of this book. We will discuss the design layout, how proofs and concepts will be described, and mathematical conventions maintained in the book. This page will have a dual focus of both informing you, the curious burgeoning mathematician, as well as you, the Wikibookians here wondering how to make your mark on this book. As such, this page will have a light-hearted fusion of both editor comments and general reader comments.
To make it easier to skim, we created three headings outlining how this wikibook ought to be designed.
* Didactic—how the content is told; what the content is
* Design—how the content looks on the medium (Wikibooks)
* Description—how the content is presented; how it reads

## Didactic

This wikibook's main headers correspond to the various concepts of elementary mathematics, the name we refer to when discussing the mathematics usually taught at high school or below. Since education mandates that elementary mathematics must be known, this wikibook uses that as a starting ground. However, you may quickly find that although it covers all concepts discussed in elementary mathematics, this wikibook also introduces many new disciplines of mathematics that one may pursue further in their higher mathematical journey. This is good! If you find a certain topic may not be explained with enough rigor, or was brushed off with an axiom, then maybe you can spend time taking a crack at it! However enjoyable it is to prove what were axioms or unexplained phenomenon, remember that some reductions back to axiomatic, informal, or "intuitive" explanations for various theorems or proofs exist for a purpose. These, however sad, are actually meant to shed light on the primary question mathematicians struggle with, and that is what mathematicians can, cannot, or are unable to work without, assuming. This is what this wikibook is meant to teach. Not only is the depth involved with every elementary mathematical concept well trodden, but how much deeper there is left to know in math. Not bad for essentially a group of abstract rules we agreed to follow, huh?

### Audience

Hello! This book is aimed towards the curious mathematicians starting out their journey of mathematics. Preferably, this mathematician should know high school mathematics. This will make most of the concepts at least familiar. However, this wikibook will look at it from a fresh new angle. Since it also contains complex topics buried inside, it is also a refreshing read for advanced mathematicians who want to clear up exactly how concepts relate to one another.

### Interpretation

How should this wikibook be read?
To begin, there are some conventions in terms of chapters:
* For now, this book is best read sequentially. Concepts build on top of each other the deeper you go.
Have you noticed Appendices or Topics you never heard of before?
* Some chapters will have an appendix heading, typically located at the bottom. These pages do not have any direct relationship to our assumed audience, since most of the topics will discuss things never-heard-of before. However, they do provide a nice primer to higher mathematics.
* In fact, some sections will be devoted to higher mathematical concepts. The first chapter will usually be a primer on what the topic is, to the uninitiated. However, the rest of the chapter assumes a level of understanding similar to your understanding of math before you read Analysis, and remember how we assumed you completed secondary education on mathematics—with some knowledge of calculus already—or understood the Calculus wikibook beforehand. It won't look easy!

### Exercises

There are problem sets in this wikibook to complement your learning!

## Design

Although this wikibook is called Real Analysis, it also introduces many other fields of mathematics, ranging from metric spaces, algebra, and set theory as well. This is the primary design focus of this wikibook, in accordance with its didactic. This book ought to introduce, to any budding mathematician, mathematics grander than simply elementary mathematics over real numbers, which quite frankly, is not defined, bounded, or really used in either the exercises or the sections until the very end of the wikibook.

### Templates

This book employs a fairly uniform template style, as well as new styles that should be implemented. Each template will be given its own row to explain itself.
| A Common Template usage | Rationale |
| {{header | title = [[../]] | author = | section = [[../Chapter_Directory|Chapter Title]] | previous = [[../Chapter_Directory_Pre|Chapter Title Pre]] | next = [[../Chapter_Directory_Nxt|Chapter Title Nxt]] | notes = }} | Headers should be on every chapter. Previous and Next sections should exists only for what is reasonably the next and previous chapter of some overall topic. |
| {{TOC|limit=3|float=right}} | Table of contents go on the right, because important links should exists on the right side like a well-designed Wikipedia page. Also the limit is set at 3 so that you can use one large heading and a subheading before time to use the proof heading, which should be hidden. |
| {{Math|size=1.2em|Content}} | Should be used for mathematical statements except when complex. They should especially be used for theorem summaries because they can be processed using a search engine. |
| {{Equation box 1}} | Should be used to summarize definitions and theorems, with the title in bold and either stating "Theorem" or "Definition of X". |

### Linking

The rationale for links on this wikibook are the same as Wikipedia's: to help the reader. Because this wikibook is on an online medium and that Wikibooks is a sister project of Wikipedia, it is the case that all pages on this wikibook should also share in Wikipedia's linking style, as mandated on their manual of style page. We also wish to prioritize links to WIkipedia's sisters projects first, before adding an external link. This is because of style reasons; external links on MediaWiki software is intentionally designed differently and with more emphasis than internal links. The focus should be on the content, not the link.
In choosing which link to add, there should be a focus on linking content in this wikibook or on equally complex wikibooks. This is out of necessity. Since most content online does not explore mathematics at a level that this book explores, providing links to more simplistic examples may not help our reader understand certain subjects. It also helps bring the advanced mathematics community together. However, this should not deter you, the editor, from adding links to other things if desired.
In summary:
* Add links like any Wikipedia page.
* Try to add complementary content on Wikibooks before adding other things.
* Be reasonable with what you link to.

### References

Since this is a sister project of Wikipedia, it is often the case that the lack of easily seen references may jar you, the audience, when you first read this wikibook. This is fair, since MediaWiki software uniformly designs the webpage to be the same. Like most books, the references are often located on a separate page, which this wikibook also contains. The link to it is here.
However, you, the editor, may wish to add new references. The rule for this is that if the reference is not global (i.e. you will not intend to use it for a good portion of the book, or the information in the reference cannot support a good portion, like a section, of this wikibook), it should be placed per page, using standard MediaWiki citation protocols. An example of how to do so can be found on Wikipedia's guide on citing sources.
In summary:
* Cite your sources.
* Citations used throughout the wikibook, by default, are located here.
* One-off citations should be done through the usual citation method.
* Use MediaWiki's citation system.

## Description
### Notation

Do not shy away from using notation! However, there should also be a way to describe your concept using words as well.

### Wikibook Terminology

This wikibook standardizes how it references itself. In a single run on sentence, the class of relations between pages are: In this wikibook, there contains several sections describing separate, but interconnected topics on higher mathematics, which each section containing several chapters that examine, in-depth, a certain concept that the section wishes to describe which, in each section, is handled usually by a single page (although a single page may be referred by more than one chapter in a given section) with multiple headings, which breaks down the topic into easy to read pieces.
Links to the referent content may be placed when the content refers to these locations.
This wikibook also draws a distinction on elementary mathematics and higher mathematics, which are defined below:
   Elementary Mathematics: the mathematics one usually learns all the way up to a high school or equivalent education. Usual concepts learns included algebraic manipulations and how to compute trigonometric functions, derivatives, and integrals. However, mathematical rigor is usually not assumed when this word is used, or a very basic understanding of it.
   Higher Mathematics: the mathematics one usually learns in a post-secondary or equivalent institution. Free online content for these materials tends to be lacking, or at the least, difficult to find. Often associated with a drive to understand what mathematics is through rigorous analysis. To gain a better feel of what that is, read the Introduction presented on the front page of this wikibook.

### Mathematical Terminology

In mathematics, there exists a set of logical operators that we use to describe our proofs. Since mathematicians are trained in using these logical notations for so long, the written equivalent of the set of logical operations tends to appear in mathematical proof and paragraphs describing concepts.
The general rules for the use of mathematical language in this book are
* They are generally fine, but introductory statements and the earlier chapters should avoid using it so our curious future mathematicians can be eased in to the discipline.
* When writing summaries of theorems, they should be written with the fact that many of our terms do not carry the necessary impact that the theorems lead on to show. Therefore, mathematically-critical concepts should be written using logical notation and everything else should be written with impact.
To clarify point 2, These words do exist. For example, the word implies is not a strong, certain word in normal parlance, even though in mathematics, the word implies is certain. So, to convey this to our readers, we can employ words like must, will, or even is to provide the necessary emphasis. However, this should not take precedence over being incorrect with your usage though. As always, simply writing out the statements with logical notation peppered in will bring your point across a lot clearer.

### Proofs

Proofs are generally more than cursory applications of definitions. As such, theorems may be used. In mathematics, mentioning theorems are akin to highlighting - they serve only to accent a series of steps like how people will not write out every algebraic step. As such, more advanced theorems in higher mathematics tend to omit or forget to state even the theorems you will learn here. These people also tend to write the textbook you will read on Real Analysis. As it is our goal to supplement, or potentially supplant, your textbook, our textbook will provided the necessary training wheels to build a solid foundation. With enough problems, you will also begin to remove those training wheels as well. This should not be done during your chapter reading.
In terms of how proofs will be formatted, proofs will
* Omit only algebraic and elementary mathematical analysis and logic.
* Expressed in a tabular format, with descriptions in the 1st column and any mathematical steps in the 2nd column.

### Conventions

* Functions should be notated as variables when possible.
* Function composition should use the ring (∘ - code &#8728;) operator for clarity when composition may be conflicted with variables.
* Functions, when typed outside of a LaTeX setting, should use the hooked HTML entity (ƒ - code &fnof;) for visual distinction.

### Appendices

On some pages, there are headings labeled "Appendix". These headings often offer alternative content which often introduce new concepts in mathematics to you, the reader. They usually related, in some sense, to the content in the page. They are optional and are not necessary to understand the purpose of this wikibook. As such, these appendices may be safely ignored when focusing on this wikibook. But be warned, not only may some questions borrow from them, but they may be important later on in your mathematical journey!