Mathematical analysis is the systematic study of real- and complex-valued functions. A large part of mathematics can be considered as mathematical analysis, including differentiation, integration, dynamical system, series and functional analysis.

Notes:

- Course Notes on Mathematical Analysis 1&2
- Notes on Mathematical Analysis 3
- Course Notes on Complex Analysis
- Functional Analysis: Special Linear Operators;

Handouts:

- Unitary Transformations in L2(I, C) - Bochner's Theorem
- Distance to a subspace not attainable
- The Kernel Method
- Open Mapping Theorem
- Closed Subspaces with Zero Aperture

**Real function** $f: \mathbb{R} \mapsto \mathbb{R}$ is a map between (subsets of) the real line.
**Real-valued function** $f: X \mapsto \mathbb{R}$ is a map whose codomain is the real line.
**Function of a real variable** $f: \mathbb{R} \mapsto X$ is a map whose domain is the real line.

**Zero set** $f^{-1}(0)$ of a real/vector-valued function is the preimage of zero.
**Sublevel set** $f^{-1}(-\infty, c]$ of a real-valued function on a set
is the preimage of an upper-bounded interval.

**Extremal point** of a continuous real function on an interval
is a point that maximizes the absolute function:
$f \in C([a, b])$, $x_0 \in \arg\max_{x \in [a, b]} |f(x)|$.
**Critical point** and **regular point** of a differentiable real-valued function
is a point where the differential is zero or nonzero, respectively.

**Partial derivative** $\frac{\partial f}{\partial x^j}(a)$
of a real-valued function on an open subset of a Euclidean space at a point
is its ordinary derivative w.r.t. a coordinate while the other coordinates are fixed:
$\frac{\partial f}{\partial x^j}(a) = \lim_{h \to 0} [f(a + h e_j) - f(a)] / h$.
**Jacobian matrix** $(\partial F^i / \partial x^j)$
of a function between (open subsets of) Euclidean spaces
is the matrix of the partial derivatives of its component functions.

**Continuously differentiable function** between (open subsets of) Euclidean spaces
is one such that all the partial derivatives of its component functions
exist and are continuous at each point in its domain: $f \in C^1(U, \mathbb{R}^m)$.
**Smooth function** between (open subsets of) Euclidean spaces
is one such that all the partial derivatives of all orders of its component functions
exist and are continuous at each point in its domain:
$\{\frac{\partial^k F^i}{\partial x^J} : k \in \mathbb{N}, J \in n^k, i \in m\} \subset C^0(U)$;
denoted as $f \in C^\infty(U, \mathbb{R}^m)$.
**Smooth function** between subsets of Euclidean spaces
is one that admits a smooth extension in a neighborhood of each point in its domain;
a global extension is unnecessary.

**Total derivative** $DF(a)$ of a map $F: U \mapsto W$
between (open subsets of) finite-dimensional normed vector spaces
that is differentiable at an interior point $a$ of its domain
is the linear map $DF(a) \in \mathcal{L}(V, W)$ that satisfies
$F(a+v) = F(a) + DF(a) v + R(v)$, $\lim_{v \to 0} |R(v)|/|v| = 0$.
The total derivative of a function between (open subsets of) Euclidean spaces
that is differentiable at an interior point of its domain
is the linear map whose standard matrix representation is the Jacobian of the map at that point:
$DF(a) = (\partial F^i / \partial x^j)(a)$.
**Directional derivative** $D_v f(a)$ of a smooth real-valued function
on an open subset of a Euclidean space at a point $a$ in the direction $v$
is the ordinary derivative of the composite function $g(t) = f(a + t v)$ at zero:
$D_v f(a) = d f(a + t v) / d t |_{t=0}$; by chain rule, we have $D_v f(a) = Df(a) v$.

Real **analytic function** (解析函数) is a real function such that at every point in the domain
there is a power series that converges to the function on a neighborhood of the point:
$\forall x_0 \in U$, $\exists (a_n)_{n \in \mathbb{N}}$, $\exists U(x_0)$:
$\sum_{n=0}^\infty a_n (x - x_0)^n = f|_{U(x_0)}$.
Real-valued **analytic function** in k variables is a real-valued function on an open subset of
the Euclidean k-space such that at every point in the domain there is a series of
homogeneous polynomials that converges to the function on a neighborhood of the point:
$\forall x_0 \in U$, $(P_n)_{n \in \mathbb{N}} \subset \mathbb{R}[x_i]_{i=1}^k$, $P_n(x_i)_{i=1}^k
= \sum_{I \in \mathbb{N}^k}^{\|I\|_1 = n} a_I \prod_{i=1}^k (x_i - x_{0,i})(x_i)^{I_i}$:
$\sum_{n=0}^\infty P_n = f|_{U(x_0)}$.
The set of all analytic functions on a set is denoted as $C^\omega(X)$.
Every analytic function is smooth: $C^\omega(X) \subset C^\infty(X)$;
however, smooth real functions may not be analytic.
**Flat function** is a smooth real function all of whose derivatives vanish at a given point.
Nonconstant flat functions are non-analytic; for example, $f(x) = e^{-1/x^2}$, $f(0) = 0$.

**Asymptotic notation**, **order notation**, or **Bachmann–Landau notation**
is a collection of notations that describe the limiting behavior of a function w.r.t. another
when the argument tends towards a particular value or infinity.
These notations first appeared in analytic number theory [@Bachmann1894; @Landau1909],
and are useful in approximation of functions, computational complexity of algorithms, and statistics.

**Big O notation** $f = \mathcal{O}(g)$ or $f = O(g)$ for functions on $X$
means that the former is absolutely bounded by a constant multiple of the latter:
$\exists c: \forall x \in X, |f| \le c |g|$.
**Little O notation** $f = o(g)$ for functions with limit point $x_0$
mean that the former is dominated by the latter asymptotically:
$\lim_{x \to x_0} f(x) / g(x) = 0$.
**Soft-O notation** $f = \tilde{\mathcal{O}}(g)$ or $f = \tilde{O}(g)$ for functions on $X$
is similar to the big O notation, but ignoring poly-logarithmic factors:
$\exists k: f = \mathcal{O}(g \log^k g)$.
This is a reasonable simplification because: $\forall k$, $\forall ε > 0$, $\log^k n = o(n^ε)$.

**Order equivalence** $f \asymp g$ or **big theta notation** $f = \Theta(g)$ for functions on $X$
means that both are absolutely bounded by a constant multiple of the other:
$f \asymp g$ iff $f = \mathcal{O}(g)$ and $g = \mathcal{O}(f)$.
**Asymptotic equivalence** $f \sim g$ for functions with limit point $x_0$
means that their ratio converges to one: $\lim_{x \to x_0} f(x) / g(x) = 1$.

**Big omega notation** $f = \Omega(g)$ and **little omega notation** $f = \omega(g)$
are the inverse of the big O notation and the little O notation, respectively:
$f = \Omega(g)$ iff $g = \mathcal{O}(f)$; $f = \omega(g)$ iff $g = o(f)$.
Knuth **big omega notation** $f = \Omega(g)$ in computational complexity theory
means the former is bounded below by the latter as the argument goes to infinity:
$\liminf_{n \to \infty} f(n) / g(n) > 0$.

**Big O in probability notation** $X_n = \mathcal{O}_p(a_n)$
means that the sequence of random variable divided by numbers is stochastically bounded:
$\forall \varepsilon > 0$, $\exists C, N > 0$:
$\forall n > N$, $P(|X_n / a_n| > C) < \varepsilon$.
**Little O in probability notation** $X_n = o_p(a_n)$
means that the sequence of random variable divided by numbers converges in probability:
$\forall \varepsilon > 0$, $\lim_{n \to \infty} P(|X_n / a_n| > \varepsilon) = 0$.

**Holomorphic function** (全纯函数) is a complex function on an open subset
that is differentiable everywhere.
A complex function is holomorphic if and only if it is analytic.
**Entire function** is a holomorphic function on the complex plane.

An **element** of an analytic function is its power series at a point
that has a non-zero radius of convergence: $W_0(z) = \sum_{n \in \mathbb{N}} a_n (z - z_0)^n$,
with disc of convergence $K_0 \supset B_r(z_0)$ and $W_0 = f|K_0$.
A **direct analytic continuation** of an element $W_0$
is an element $W_1$ at another point in $W_0$'s disc of convergence:
$W_1(z) = \sum_{n \in \mathbb{N}} b_n (z - z_1)^n$, where $z_1 \in K_0$, $z_1 \ne z_0$.
An **analytic continuation** of an element $W_0$
is an element $W_n$ obtained by a chain of direct analytic continuations.

The **complete analytic function** generated by an element $W_0$ is
the union of all analytic continuations of $W_0$:
$W = {W_n : n \in \mathbb{N}, \forall i \in n, z_{i+1} \in K_i}$.
The **domain of existence** of a complete function
is the union of the discs of convergence of its elements: $D = \bigcup{K_n}$.
A complete analytic function is generally a multi-valued function on $D$.
A **branch** of an analytic function,
given a domain $D' \subset D$ and an element $W_0$ centered in $D'$,
is the union of all elements that can be obtained by analytic continuation of $W_0$
by means of chains with centres belonging to $D'$:
$W' = {W_n : n \in \mathbb{N}, \forall i \in n, z_{i+1} \in K_i, z_n \in D'}$.
If $D'$ is simply connected, the branch is single-valued in $D'$.
The n-th root $\sqrt[n]{z}$ has exactly $n$ single-valued branches;
The logarithm $\ln(z)$ has infinitely many single-valued branches.

A **singular point** of an analytic function is an obstacle to the analytic continuation
of an element along any curve of a specific class in the z-plane.
A **regular point** is a point that is not singular.
An **isolated singular point** is a singular point $a$ such that
analytic continuation is possible to all points sufficiently close to $a$.
A **single-valued singular point** is an isolated singular point
such that analytic continuations along closed curves are identical.
A **pole** is a single-valued singular point if an infinite limit exists: $\lim f(z) = \infty$.
An **essential singular point** is a single-valued singular point that is not a pole.
A **branch point** or a **many-valued singular point**
is an isolated singular point that is not single-valued.
An **algebraic branch point** of order $m$, $m \ge 1$, is a branch point $a$ such that
analytic continuations along $m + 1$ single loops around $a$ in the same direction are identical.
A **transcendental branch point** is a branch point that is not algebraic.

Vector calculus:

Curvilinear coordinates:

- [@Lax1966] Functional Analysis;
- [@Kolmogorov and Fomin, 1970] Introductory Real Analysis;
- [@Kreyszig1978] Introductory Functional Analysis with Applications;
- [@Paulsen and Raghupathi, 2016] An Introduction to the Theory of Reproducing Kernel Hilbert Spaces;