Real Analysis¶

Job: rigorous calculus on $\mathbb{R}^n$ — completeness, limits, differentiation, integration, and interchange theorems used as baseline for research analysis.

Core¶

$\mathbb{R}$ as complete ordered field. Supremum property $\Leftrightarrow$ Cauchy completeness $\Leftrightarrow$ nested interval property. $\mathbb{Q}$ dense; $\mathbb{R}\setminus\mathbb{Q}$ dense. Countable $\Leftrightarrow$ bijection with $\mathbb{N}$; $\mathbb{R}$ uncountable (Cantor).

Topic	Core facts
Sequences	limsup/liminf; Bolzano–Weierstrass: bounded $\Rightarrow$ convergent subsequence
Series	absolute $\Rightarrow$ convergent; conditional (alternating); $\sum 1/n^p$
Topology of $\mathbb{R}^n$	all norms equivalent; compact $\Leftrightarrow$ closed bounded
Uniform continuity	continuous on compact $\Rightarrow$ uniformly continuous
Differentiation	mean value thm; Taylor with remainder; inverse/implicit function thms
Riemann integral	continuous on $[a,b]$ integrable; FTC
Uniform convergence	preserves continuity; $\int$ limit = limit $\int$ under unif; Weierstrass $M$-test

Differentiation ($\mathbb{R}^n$)¶

$f:U\subseteq\mathbb{R}^n\to\mathbb{R}^m$ differentiable at $a$ if $\exists\,Df(a)\in L(\mathbb{R}^n,\mathbb{R}^m)$ with $$ \lim_{h\to 0}\frac{|f(a+h)-f(a)-Df(a)h|}{|h|}=0. $$ $C^1$ $\Rightarrow$ differentiable; existence of partials alone insufficient (standard counterexamples).

Theorem (Inverse function). $f\in C^1$, $Df(a)$ invertible $\Rightarrow$ $f$ local $C^1$-diffeomorphism.

Theorem (Implicit function). $F(x,y)=0$, $D_y F(a,b)$ invertible $\Rightarrow$ local $y=g(x)$ with $F(x,g(x))=0$.

Integration and interchange¶

Riemann vs Lebesgue (see measure theory): Lebesgue handles $L^p$ limits, Fubini, dominated convergence.

Uniform convergence pitfalls. Pointwise limit of continuous can be discontinuous; of differentiable need not be differentiable; termwise differentiation requires extra hypotheses (e.g. uniform conv of derivatives + conv at a point).

Series of functions¶

Power series: radius of convergence; $C^\infty$ on interval of conv; not always analytic ($e^{-1/x^2}$). Fourier series: $L^2$ convergence; pointwise requires more (Carleson for $L^2$; classical Dini/ Dirichlet tests).

Multivariable integral theorems (calc interface)¶

Green / Stokes / divergence: $\int_M d\omega=\int_{\partial M}\omega$ in differential form language — precise under smooth compact oriented manifolds with boundary (or Lipschitz domains with care).

See it¶

Implicit function theorem in moduli / constraints. Solving $F(x,\lambda)=0$ for bifurcation: $D_x F$ Fredholm of index $0$, Lyapunov–Schmidt reduction — finite-dimensional IFT is the local model.

Weierstrass approximation / Stone–Weierstrass. Polynomials dense in $C[a,b]$ — justifies spectral theory approximations and Stone–Čech-free arguments in compact Hausdorff spaces with subalgebras.

Check¶

Give a sequence of continuous $f_n\to f$ pointwise with $f$ discontinuous, and explain why convergence is not uniform.

$f_n(x)=x^n$ on $[0,1]$: pointwise limit $0$ on $[0,1)$, $1$ at $1$. Uniform limit of continuous is continuous, so convergence not uniform on $[0,1]$.

Why does existence of all directional derivatives not imply Gâteaux/Fréchet differentiability?

Directional derivatives may exist but fail linearity in the direction, or the limit defining the derivative may fail to be uniform on the unit sphere (Fréchet). Classic examples on $\mathbb{R}^2$ with partials everywhere but not continuous and not complex-differentiable / not $\mathbb{R}$-linear differential.

Pitfalls¶

Interchanging lim and $\int$ / $\sum$ / $\partial$ requires domination, uniformity, or monotone convergence — pointwise alone fails.
$C^\infty\neq$ analytic on $\mathbb{R}$.
Pathological functions (using AC): nowhere continuous, non-measurable — measure theory draws the usable line.
Norm equivalence fails in infinite dimensions — real analysis finite-dim facts do not export to Banach spaces without proof.

Refs¶

Rudin, Principles of Mathematical Analysis
Apostol, Mathematical Analysis
Spivak, Calculus on Manifolds (IFT / Stokes package)

Next¶

Metric spaces · Measure theory · Functional analysis · Complex analysis · Topology overview