# Functional analysis and real analysis

• Gauss M.D.
In summary, the conversation discusses the importance of taking real and functional analysis courses for those interested in a career in applied mathematics. It is mentioned that taking real analysis may be beneficial as a prerequisite for functional analysis, but it ultimately depends on the details of each course and program. Some individuals believe that both courses are essential, while others suggest focusing on more practical courses such as programming or algebra. Ultimately, it is advised to discuss this with a math adviser and consider personal strengths and priorities before enrolling in these courses.

#### Gauss M.D.

In my schools functional analysis course, under prerequisites, it says "real analysis would be a good preparatory course, but is not required". In the concurrent real analysis thread, it was mentioned that real analysis is a stepping stone to functional analysis.

1) how essential is taking real analysis prior to functional analysis?
2) how essential is functional analysis to someone interested in computational science, simulation and general mathematical modeling (including statistic/stochastic modelling)?

Until now, I've assumed both courses to be no-brainers for people interested in a career in applied math, but if both real and functional analysis are abstract and proof driven, bordering on "pure math", maybe ones focus should be elsewhere.

Given infinite time, I'd be happy to take both courses. But if they're competing with programming courses, advanced linear algebra/numerical analysis, fluid dynamics, quantum mechanics etc, are they really worth it?

You need to discuss this with your math adviser - it depends on the details of each course as taught.

In many programs functional analysis assumes that you have had the senior level real analysis - in other programs they may only require an "introduction to real analysis".

More linear algebra is always good for physics! And today everybody needs to know programming - and for physics a course in numerical analysis is more useful than real analysis.

I took this class and it was one of the biggest mistakes I made,

It was kind of interesting, but I have never used ANYTHING besides the definition of Hilbert spaces and metrics and stuff like that ever again. More importantly, while I was taking the class I complained about it to a bunch of physics academics and none of them had taken this kind of class before, if they ever had to use it they just learned what they needed, which is what I would have preferred.

In my opinion this subject isn't something you can quickly learn in one semester while juggling several other classes, it's something that requires A LOT of time and basically, unless you're a genius, it requires you do a lot of example questions to grasp the concepts.

I'm not really sure why or where it is used in programming, but I am not an experienced programmer. Where I'm from the only degree's that suggest real and functional analysis are actuarial studies, statistics and mathematics.

If I were you I'd pick a programming course or like UltrafastPED said, the algebra class.

I hope I haven't replied too late.

Both subjects are pure mathematics, and almost completely useless to you. You're better off simply picking up useful tidbits on the go than risking your GPA and wasting your time.

Now, if you are extremely talented and enjoy proof based pure mathematics, and do not think either course will pose a threat to your grades/research/whatever other important things you're up to, by all means try it.

I understand your concerns about the importance of taking real analysis and functional analysis courses in relation to your career goals in computational science, simulation, and mathematical modeling. Here are my thoughts on your questions:

1) It is generally recommended to have a solid understanding of real analysis before diving into functional analysis. Real analysis provides the foundation for many concepts and techniques used in functional analysis, such as topology, convergence, and continuity. Without this background, it may be difficult to fully grasp the material in functional analysis. However, it is not impossible to learn functional analysis without real analysis, especially if you have a strong mathematical background and are willing to put in extra effort to fill any gaps in your knowledge.

2) Functional analysis is an important tool in many areas of applied math, including computational science, simulation, and modeling. It allows for a more abstract and general approach to problem-solving, which can be very useful when dealing with complex systems. It also has applications in statistics and stochastic modeling, as many statistical methods are based on functional analysis principles. Therefore, I would say that functional analysis is definitely worth your time and effort if you are interested in pursuing a career in these fields.

However, as you mentioned, there are also other courses that may be more directly applicable to your career goals. It ultimately depends on your specific interests and the requirements of your field. I would recommend researching the specific topics and techniques covered in both real and functional analysis courses offered at your school, and then deciding which one aligns more closely with your interests and career goals. Remember, having a diverse and well-rounded mathematical background can also be beneficial in the long run.

In conclusion, while real and functional analysis may seem like abstract and proof-driven courses, they can be valuable tools for applied mathematicians. It is ultimately up to you to weigh the importance of these courses in relation to your other academic and career pursuits.

## 1. What is the difference between functional analysis and real analysis?

Functional analysis is a branch of mathematics that deals with infinite-dimensional vector spaces and their operators, while real analysis focuses on the study of real numbers and their properties. In functional analysis, the objects of study are usually functions, while in real analysis, the focus is on numbers and their behavior.

## 2. What are some applications of functional analysis?

Functional analysis has many applications in different fields, such as physics, engineering, economics, and computer science. In physics, it is used to study quantum mechanics and general relativity. In engineering, it is applied to control theory and signal processing. In economics, it is used in game theory and optimization. In computer science, it is used in data compression and image processing.

## 3. What are the basic concepts in functional analysis?

Some of the basic concepts in functional analysis include vector spaces, linear operators, norms, inner products, and dual spaces. Vector spaces are sets of objects (usually functions) that can be added and multiplied by scalars. Linear operators are functions that map one vector space to another. Norms are measures of the size of a vector. Inner products are a generalization of the dot product in real analysis. Dual spaces are spaces of linear functionals, which are functions that map vectors to scalars.

## 4. How does functional analysis relate to other branches of mathematics?

Functional analysis has connections with many other branches of mathematics, such as topology, differential equations, and measure theory. The study of topological vector spaces is a combination of functional analysis and topology. The concept of a functional, which is a linear map from a vector space to its underlying field, is also used in differential equations. Measure theory is used in functional analysis to define measures on infinite-dimensional spaces.

## 5. What are some important theorems in functional analysis?

There are many important theorems in functional analysis, such as the Hahn-Banach theorem, the Banach-Steinhaus theorem, and the open mapping theorem. The Hahn-Banach theorem states that every continuous linear functional on a subspace can be extended to the whole space. The Banach-Steinhaus theorem, also known as the uniform boundedness principle, states that if a family of linear operators is pointwise bounded, then it is uniformly bounded. The open mapping theorem states that if a linear operator between Banach spaces is surjective, then it is an open map.