Part I

One way and another, I got a good
background in real analysis. So, okay,
I'll try to answer:

An answer depends on what is meant by
real analysis.

Part of the answer is advanced calculus,
and part of that is the Gauss, Green, and
Stokes theorems. If do these the modern
and high end ways, then get as deep as
like, and spend as much time as like, in
differential geometry, calculus on
manifolds, differential forms, E. Cartan,
exterior algebra, algebraic topology, etc.
But for 19th century physics and
engineering, there is a way to get a good
treatment of what need in about a nice
weekend from (own TeX markup):

Tom M.\ Apostol, {\it Mathematical
Analysis: A Modern Approach to Advanced
Calculus,\/} Addison-Wesley, Reading,
Massachusetts, 1957.\ \

Note how old it is! It's no longer really
"modern"! Get a used copy; that's what I
did. You don't want a newer edition!

But with that out of the way, might take a
fast pass through an old MIT standard

Francis B.\ Hildebrand, {\it Advanced
Calculus for Applications,\/}
Prentice-Hall, Englewood Cliffs, NJ,
1962.\ \

There's a lot of fun stuff in there, but
it's TOO short on proofs. So, I'd pass it
up and return later when know enough real
analysis to guess or understand most of
the proofs easily.

Note: In (the relatively elegant and easy
to read)

George F.\ Simmons, {\it Introduction to
Topology and Modern Analysis,\/} McGraw
Hill, New York, 1963.\ \

the flat statement IIRC "The two pillars
of analysis are linearity and continuity."

Well, for the linearity, really need a
good background in linear algebra. For
this, you need at least three books, a
really easy one, elementary, that starts
with, say, systems of linear equations and
Gauss elimination. Then a more advanced
one that emphasizes the axioms for a
vector space, does vector spaces with at
least both of the real and complex numbers
(over finite fields can be important for
computing but not for real analysis) and
emphasizes eigenvalues and eigenvectors,
and finally the grand one, written at the
knee of von Neumann as a finite
dimensional introduction to Hilbert space
theory,

Paul R.\ Halmos, {\it Finite-Dimensional
Vector Spaces, Second Edition,\/} D.\ Van
Nostrand Company, Inc., Princeton, New
Jersey, 1958.\ \

If you get very far in real analysis,
then you will want a good treatment of at
least basic Hilbert space theory, and
Halmos is a good start.

For his section on multi-linear algebra,
I'd skip that unless plan to take the
exterior algebra of differential forms
seriously. In that Halmos book,
concentrate on vector spaces, vector
subspaces, linear transformations,
eigenvalues, eigenvectors, and Hermitian
and unitary matrices. Also at the end
note the cute ergodic theorem! The big
deal in about the last half of the book is
spectral decomposition -- don't skip that.

For your second book, I used E. Nearing --
he was a student of E. Artin at Princeton.
So, Nearing's book is high quality stuff.
I worked carefully through that and
learned a lot. But his appendix on linear
programming is a disaster! Can do nearly
everything important in linear programming
and its simplex algorithm as just a simple
-- learn it in an hour -- extension of
Gauss elimination. Nearing does finite
cones and dual cones for which he never makes
a clear connection with linear
programming. And although he works with
all those cones, still he misses the
theorems of the alternative -- Farkas,
etc. -- important in parts of
optimization, convexity, etc.

Also recommended is Hoffman and Kunze and,
IIRC, available for free on the Internet
as a PDF file.

There is much more in linear algebra,
e.g., from R. Bellman, R. Horn, on
numerical methods, etc. but these are not
crucial for a rush to real analysis.

Maybe part of real analysis, that is,
advanced enough, is "Baby Rudin",
Principles of Mathematical Analysis.
The later editions have near the end some
tacked on material, mostly without
sufficient context, on the exterior
algebra of differential forms. Skip that.
If you want that material, then go for R.
Buck, Advanced Calculus or Spivak,
Calculus on Manifolds or really just go
for a real book on differential geometry,
manifolds, calculus on manifolds, etc.
Such differential geometry is from
important to crucial for several
objectives but is NOT on the mainline of a
rush to real analysis.

So, what is going on in Baby Rudin? Okay,
the main idea of the book is that we can
give a solid development of the Riemann
integral for a function that is continuous
on a compact set. So, get to learn
about continuity, that is, one of the two
pillars of analysis. Then hand in hand
with continuity is compactness, so get to
see that. All of that is in just the
first few chapters; that's what those
first few chapters are all about -- continuity
and compactness. E.g., get to learn that
in R^n (for the set of real numbers R and
a positive integer n), a set is compact
if an only it is closed and bounded --
super, important, crucial stuff, the
key to a clean up of Riemann integration,
that is, material Newton didn't know.

Then with that material on continuity
and compactness, Rudin
does the
Riemann–Stieltjes integral. There, mostly
just ignore the Stieltjes part with its
possibility of step functions (maybe as
a cheap answer to what the physics people
try to do with the Dirac delta function,
which, of course, is not really a
function, but has a clean fix-up with
distributions and measure theory) and just
read that Rudin material for the Riemann integral of
first calculus. You will likely never see
the Stieltjes extension again.

The main idea is: A function continuous
on a compact set is also, presto, bingo,
wonder of wonders, really nice day,
uniformly continuous, and that makes the
derivation of the Riemann integral really
easy. Really, that's the core idea of the
whole book. Baby Rudin can seem severe,
but with this introduction you should be
able actually to like it a lot. Later in
the book, Rudin touches on the fact that
the uniform limit of a sequence of
continuous functions is continuous -- same
song, next verse. It was a Ph.D.
qualifying exam question for me; I did get
it.

Later he gives a really nice treatment of
Fourier series -- that is very much worth
reading.

I'd suggest one side trip: Cover the
inverse and implicit function theorems.
They are just a local, non-linear
generalization of what you will see really
easily for linear transformations in
linear algebra via, right, just Gauss
elimination. For a source? There is a
good treatment in W. Fleming, Functions
of Several Variables. IIRC, there is also
a cute proof based on contractive mappings.

So, by then you will have a good start on both
continuity and linearity.