When reading the prefaces of many books devoted to the theory of inequalities, I found one thing repeatedly stated: Inequalities are used in all branches of mathematics. But seriously, how important are they? Having finished a standard freshman course in calculus, I have hardly ever used even the most renowned inequalities like the Cauchy-Schwarz inequality. I know that this is due to the fact that I have not yet delved into the field of more advanced mathematics, so I would like to know just how important they are. While these inequalities are usually concerned with a finite set of numbers, I guess they must be generalised to fit into subjects like analysis. Can you provide some examples to illustrate how inequalities are used in more advanced mathematics?
[Math] How important are inequalities
inequalitysoft-question
Related Solutions
To add a remark related to Jim Belk's answer and the OP's comments on that answer:
In many naturally occurring situations, including some of those where group theory is particularly useful, endomorphisms are automatically automorphisms.
For example, if $E/F$ is a finite extension of fields, any endomorphism of $E$ which is the identity on $F$ is automatically an automorphism of $E$.
As another example, if $C$ is a Riemann surface of genus at least $2$, then any (nonconstant) endomorphism of $C$ is necessarily an automorphism.
Any endomorphism of a Euclidean space which preserves lengths is necessarily an automorphism.
Another point to bear in mind is that the groups that arise in practice in geometry are often Lie groups (i.e. have a compatible topological, even smooth manifold, structure). One can define a more general notion of Lie semigroup, but if your Lie semigroup has an identity (so is a Lie monoid) and the semigroup structure is non-degenerate in some n.h. of the identity, then Lie semigroup will automatically be a Lie group (at least in a n.h. of the identity). A related remark: in the definition of a formal group, there is no need to include an explicit axiom about the existence of inverses.
To make a point related to Qiaochu Yuan's answer: in some contexts semigroups do appear naturally.
For example, studying the rings of endomorphisms of an object is a very common technique in lots of areas of mathematics. (E.g., just to make a connection to my first point, for genus $1$ Riemann surfaces, there can be endomorphisms that aren't automorphisms, but then genus $1$ Riemann surfaces can also be naturally made into abelian groups --- so-called elliptic curves --- and there is a whole theory, the theory of complex multiplication, devoted to studying their endomorphisms rings.)
As another example, any ring of char. $p > 0$ has a Frobenius endomorphism, which is not an automorphism in general; but the semigroup of endomorphisms that it generates is typically an important thing to consider in char. $p$ algebra and geometry. (Of course, this semigroup is just a quotient of $\mathbb N$.)
One thing to bear in mind is what you hope to achieve by considering the group/semigroup of automorphisms/endomorphisms.
A typical advantage of groups is that they admit a surprisingly rigid theory (e.g. semisimple Lie groups can be completely classified; finite simple groups can be completely classified), and so if you discover a group lurking in your particular mathematical context, it might be an already well-known object, or at least there might be a lot of known theory that you can apply to it to obtain greater insight into your particular situation.
Semigroups are much less rigid, and there is often correspondingly less that can be leveraged out of discovering a semigroup lurking in your particular context. But this is not always true; rings are certainly well-studied, and the appearance of a given ring in some context can often be leveraged to much advantage.
A dynamical system involving just one process can be thought of as an action of the semigroup $\mathbb N$. Here there is not that much to be obtained from the general theory of semigroups, but this is a frequently studied context. (Just to give a perhaps non-standard example, the Frobenius endomorphism of a char. $p$ ring is such a dynamical system.) But, in such contexts, precisely because general semigroup theory doesn't help much, the tools used will be different.
E.g. in topology, the Lefschetz fixed point theorem is a typical tool that is used to study an endomorphism of (i.e. discrete dynamical system on) a topological space. Interestingly, the same formula is used to study the action of Frobenius in char. $p$ geometry (see the Weil conjectures). So even in contexts such as action of the semigroup $\mathbb N$, there is some coherent philosophy that can be discerned --- it is just that it is supplied by topology rather than algebra, since the algebra doesn't have all that much to say.
I think the conclusion to be drawn is not to be too doctrinaire, and to be sensitive to the actual mathematical contexts in which and from which the various notions of group, semigroup, automorphism, and endomorphism arise and have arisen.
In going from high-school to, say, graduate${}^\color{Blue}\dagger$ level math, the higher math being "nowhere close to the math that appealed to" you is probably a very real threat. However, if your pleasure in learning analysis and algebra is any indication - instead, it will be math that you love even more.
A few things that I've noticed changing as I've learned more math are:
- Generality: Everyone knows about integers, rationals, reals, maybe complex numbers, but the next step up, conceptually, is to look at rings and fields and then modules and so forth. In higher math, the generality of our constructs increases a lot. Often we then narrow our focus again and end up looking at 'cousins' of the things we were originally studying. Other times we run into problems and it becomes the question of the decade precisely how to successfully go about a particular campaign of generalization and overcome the relevant obstacles.
- Branching: What a lot of people don't understand is that mathematics isn't linear, and doesn't progress in a rigid sequence. It branches out into many different areas, and in exploring these branches they can "feel" radically different. You can have a crush on one branch while hating another branch. In some cases, you have a love-hate relationship, or "it's complicated," etc.
- Reinterpretation: With a little bit of dabbling in different areas of math, it's possible that a single problem/idea can be attacked/framed from many different angles, using very different concepts. Sometimes this can seem "natural" and easily "motivated," while other times alien and bizarre. Frequently this sort of thing is a bit of a pastime for some mathematicians.
- Richness: In summary, mathematics becomes richer. In scaling conceptual mountains, we build concept on top of concept on top of seventeen more concepts until we're left studying situations that are saturated with structure, and when we hike back down the other side we come across the exotic or pathological; in branching we discover a high degree of diversity we hadn't previously imagined, each with comparable feel and texture to them; and then when we study wide and far we find that even our familiar notions have multiple sides to them.
$~~ {}^\color{Blue}\dagger$Yeah, disclaimer: I'm not actually there yet. :-)
Best Answer
Inequalities are extremely useful in mathematics, especially when we deal with quantities that we do not know exactly what they equate too. For example, let $p_n$ be the $n$-th prime number. We have no nice formula for $p_n$. However, we do know that $p_n \leq 2^n$. Often, one can solve a mathematical problem, by estimating an answer, rather than writing down exactly what it is. This is one way inequalities are very useful.