Lecture 1: Basic Banach Space Theory

MIT 18.102 Introduction to Functional Analysis, Spring 2021
Instructor: Dr. Casey Rodriguez

An introduction to Banach space theory, including vector spaces, norms and important examples of normed spaces.

License: Creative Commons BY-NC-SA

[mitocw]

MIT 18.102 Introduction to Functional Analysis, Spring 2021
Instructor: Dr. Casey Rodriguez

[akrishna]

Timestamps and Summary of Lecture 1: Basic Banach Space Theory

0:00 - Motivation and Introduction
Functional analysis works with vector spaces that are sometimes infinite-dimensional, using the techniques of analysis to study their structure & functions defined on them. Besides existing as a pure mathematical discipline, it finds applications in partial differential equations and physics.

5:27 - Review of Vector Spaces
Vector spaces are defined over a field (here, R or C) and are closed under addition and scalar multiplication. Every vector space has a basis, or a maximal spanning set; its dimension is the cardinality of any basis (this is well-defined). e.g. Finite-dimensional vector spaces over R: R^n (n-fold Cartesian product)

13:46 - An Infinite-Dimensional Vector Space
The set of continuous functions from [0, 1] to the complex numbers forms a vector space (sum, multiple of any elements are still in the VS). But the countably infinite set {1, x, x^2, ...} is linearly independent in C([0, 1]).

16:26 - Norms
To perform analysis, we require a concrete notion of "proximity" or "size" on our spaces. A norm assigns to each vector in a vector space some nonnegative real number; as a map it is positive semidefinite, homogeneous, and satisfies the triangle inequality.
A seminorm is similar to (but weaker than) a norm; it satisfies the latter two conditions but is not necessarily definite. A vector space endowed with a norm is a normed vector space: these are central objects of study in functional analysis.

21:15 - Metric Spaces
A metric on a set provides a notion of distance between its points. It is an identity-indiscernible, symmetric, and triangle inequality-satisfying function; a space with such a distance function is called a metric space (has the metric topology)

23:08 - Norms Induce Metrics
Given a norm on a vector space, it induces a metric in the natural way. This allows us to talk about ideas like convergence, completeness, etc. with regards to normed vector spaces.

26:11 - Examples of Norms (and NVS)
Euclidean norm on R^n, C^n provide us with the most familiar notion of distance. We can consider more broadly the family of norms called p-norms, of which the Euclidean norm is a special case (taking p = 2). When p = ∞ we consider the norm that picks out the maximal component of a vector in n-space.

32:45 - The Space C_∞{X}
From a metric space X we consider the space C_∞{X} of all continuous, bounded functions from X to the complex numbers. This indeed forms a vector space - on it, we can introduce the supremum norm that measures the maximal value a function takes. This turns C_∞(X) into a normed vector space. If we take X to be some compact interval like [0, 1], the boundedness of continuous functions are automatic.

39:24 - The Supremum Norm and Uniform Convergence
When asking what convergence in this metric means for functions in C_∞(X), we find it translates to uniform convergence in the familiar sense from real analysis.

42:20 - l^p-Spaces
The l^p spaces consist of p-summable sequences. When p = 2, these are square-summable, etc. and these are all defined to be the sequences on which the respective p-norms resolve finitely.

46:38 - Banach Spaces
We are interested in special cases of normed vector spaces that mimic the situation/structure in Euclidean spaces, namely, their completeness. We have seen that norms (on vector spaces) give rise to metrics, with respect to which we make sense of convergence of sequences. Cauchy sequences are those whose terms tend arbitrarily close; every convergent sequence is Cauchy, but the converse doesn't necessarily hold (the rationals have many "holes": one can construct a sequence of rationals that close in on sqrt(2) but can't converge to it in Q). Spaces for which the converse does hold, i.e. Cauchy sequences converge, are complete with respect to that given metric. Banach spaces are normed vector spaces that are complete with respect to the metric induced by the norm.

49:52 - Examples of Banach Spaces
R^n, C^n are Banach spaces with respect to any of the aforementioned p-norms. These provide relatively trivial examples, but foundational ones.

50:39 - C_∞(X) is Banach
A useful nontrivial example - the normed vector space of continuous, bounded functions X -> C actually forms a Banach space with respect to the supremum metric. The process of showing that it is a Banach space amounts to exhibiting completeness, and is instructive in demonstrating the general procedure for showing that a space is Banach: first take a Cauchy sequence & come up with a candidate for its limit, then show that this proposed limit lies in the space, and finally show that the convergence does occur.

[haldanesghost]

I’ve needed this course for a couple of years now. Looking for good open resources regarding such an advanced topic was hard. The search is over. Thanks OCW! You’re da GOATs of open learning.

[mikeCavalle]

the sound of hard chalk on thick slate is marvelous

[DLRhodes]

I had no idea Anthony Fantano was a functional analysis professor at MIT

[akrishna]

Thank you so much OCW & Dr. Rodriguez for putting video lectures to these notes! Since last year's 18.102 notes were published to the website, they have been the most valuable and easily-approachable resource for me when learning functional analysis.

[daniellan]

Thank you, MIT OCW team! Thank you, Dr. Rodriguez! These videos will change a lot of people's lives. It's a great contribution to the education!

[TrepidDestiny]

"You've taken linear algebra. You've taken calculus." Buddy, not even close. But please, continue! >.<

[shafqat]

Dr Rodriguez is an absolute Top G. My man dropped the Real Analysis course (which I'm still devouring) a few months ago.

This was long awaited.

Thank Prof!

[JaGWiREE]

Awesome to see a functional analysis series uploaded.

[nicolasg.b.]

Hey self-learners! At 45:11, it should be p = 1, not p = ∞.

[agcouper]

Ah, the functional analysis. Together with topology, my favorite subjects in uni.

[snk-js]

I am freestyle developer in Brazil and since started my career did never have contact with computer science academics lectures, but after some years of experience I am loving all the free lectures MIT is providing in youtube. I have watched almost the whole channel since then, and I loved this one too. I would love now to set up a goal to study in MIT personally, but don't even know how much is the financial and general requirements to get into CS grad program.

[martinsoos]

There is a reason that the ground goes down. If anyone else saw the electric outlets on the wall, the thumb doesn't contact any of the bare connectors but the finger below the thumb does and many people get electrocuted unless the ground is down.

[dedekindcuts]

Wow!! Thanks very much MIT!! Really appreciate the amazing sharing of resources and hopefully one day we will get open courses on the key undergraduate and graduate math subjects!

[freeeagle]

I will follow him
Follow him wherever he may go
There isn't an ocean too deep
A mountain so high it can keep me away

Millions of thanks to Dr. Casey Rodriguez who makes maths courses accessible, ocean not too deep and mountain not too high any more.

[randalllionelkharkrang]

Im from a Machine Learning background(masters at the moment) and you are really good and smoothly going from concept to concept, in my opinion. I hope to continue this series during the holiday season.

[BAMBAMBAMBAMBAMval]

Did one of your students build the motiontracking camera?

[mastershooter]

More functional analysis courses please!! more advanced functional analysis courses! perhaps even non-linear functional analysis :D

[ikechukwumichael]

Thank you for this opportunity to learn such a valuable material