Notation for the selected prerequisite proofs

These definitions make the selected proofs from Jiří Lebl's Basic Analysis I–II, version 6.3, readable together. They use the same conventions as the free author edition. The accompanying programme proof map identifies the complete proofs and the local completions of used exercises. A definition here does not assert a theorem about the object defined.

Ordered numbers, sequences and series

We use the ordered real field with the least-upper-bound axiom and natural-number induction. An upper bound for a set $A\subset\mathbb R$ is a number at least every member of $A$; $\sup A$ is its least upper bound when it exists. Lower bounds and $\inf A$ are defined with the order reversed. The real-field axiom guarantees $\sup A$ for every nonempty bounded-above set. We set $\inf A=-\sup(-A)$ when the right side is defined. The empty finite sum is zero and the empty finite product is one.

A sequence $x_n$ converges to $x$ if, for each $\varepsilon>0$, there is $N$ such that $|x_n-x|<\varepsilon$ for all $n\geq N$. It is bounded when $|x_n|\leq M$ for some $M$; increasing means $x_n\leq x_{n+1}$, and decreasing means the reverse inequality. A subsequence is $x_{n_j}$ with strictly increasing natural indices $n_j$. A tail discards finitely many initial terms. A real Cauchy sequence satisfies $|x_n-x_m|<\varepsilon$ for all sufficiently large $n,m$. For bounded real sequences the tail suprema and infima are $a_n=\sup_{k\geq n}x_k$, $b_n=\inf_{k\geq n}x_k$; their limits, when they exist, are $\limsup x_n$ and $\liminf x_n$.

A series is the limit of its finite partial sums when that limit exists. Absolute convergence means convergence of the series of absolute values. For nonnegative terms we also allow the value $+\infty$, defined as the supremum of finite partial sums. Complex numbers are pairs of reals with $i^2=-1$; $\overline{x+iy}=x-iy$ and $|x+iy|=\sqrt{x^2+y^2}$. The complex arithmetic and series operations used in the selected proofs are justified in the exponential companion.

Metric spaces and their topology

A metric $d$ on $X$ is nonnegative, symmetric, zero exactly for equal points, and satisfies the triangle inequality. The subspace metric is the restriction to a subset. We write $B(x,r)=\{y:d(x,y)<r\}$ and $C(x,r)=\{y:d(x,y)\leq r\}$. A subset is bounded if it lies in some ball of finite radius. In $\mathbb R^n$ the Euclidean scalar product is $x\cdot y=\sum_jx_jy_j$, the norm is $|x|=\sqrt{x\cdot x}$, and the distance is $|x-y|$.

A set is open if each of its points has a ball inside it, and closed if its complement is open. The interior consists of the points with such a ball. The closure is the intersection of the closed sets containing the set. The boundary is the closure minus the interior. A neighborhood of a point contains an open set containing that point. Relative openness uses the subspace metric. A nonempty space is connected if its only subsets that are both open and closed are itself and the empty set.

Metric convergence means $d(x_n,p)\to0$. The metric Cauchy condition uses $d(x_n,x_m)<\varepsilon$ eventually for all $n,m$. A metric space is complete if every Cauchy sequence converges in it. A set is compact if every cover by open sets has a finite subcover.

Continuity of $f:X\to Y$ at $p$ means that for each $\varepsilon>0$ there is $\delta>0$ such that $d_X(x,p)<\delta$ implies $d_Y(f(x),f(p))<\varepsilon$. Uniform continuity uses one $\delta$ for all pairs of points. A cluster point of $S$ has another point of $S$ in every ball about it. A function limit uses the same inequality on $S\setminus\{p\}$. Restrictions are denoted $f|_S$; one-sided limits restrict the domain to the corresponding half-line. A $k$-Lipschitz map satisfies $d_Y(f(x),f(y))\leq k\,d_X(x,y)$. A contraction has such a constant with $0\leq k<1$. A fixed point satisfies $f(x)=x$.

Linear algebra and differentiation

A real vector space has associative commutative addition, zero and additive inverses, and scalar multiplication satisfying the distributive, associative and unit axioms. Linear combinations are finite sums of scalar multiples. The span of a set consists of its finite linear combinations. A finite list is linearly independent if a linear combination equal to zero has all coefficients zero. A basis is an independent spanning list. The dimension is the size of a basis; the selected basis theorems justify that this does not depend on the basis.

A linear map preserves addition and scalar multiplication. $L(X,Y)$ denotes the vector space of such maps; $\ker A=\{x:Ax=0\}$, and $\operatorname{ran}A=\{Ax:x\in X\}$. An invertible map is bijective. $GL(X)$ is the set of invertible linear maps $X\to X$. Once bases are fixed, the $j$-th matrix column is the coordinate vector of the image of the $j$-th basis vector. The identity matrix is $I$, and the transpose is $A^T$.

A norm is nonnegative, zero only at zero, absolutely homogeneous, and subadditive. For a linear map its operator norm is $\|A\|=\sup_{\|x\|=1}\|Ax\|$, interpreted with the zero-space convention in the local differential companion. In the source, $\|\cdot\|$ may denote either a vector norm or its induced operator norm. The determinant is the alternating permutation sum $\det A=\sum_\sigma\operatorname{sgn}(\sigma)\prod_i a_{i,\sigma(i)}$. The parity and all determinant identities used are proved in the selected proofs and P7. A set is convex if it contains every segment joining its points.

For an open set $U\subset\mathbb R^n$, a map $f:U\to\mathbb R^m$ is differentiable at $x$ if there is a linear map $A$ such that $|f(x+h)-f(x)-Ah|/|h|\to0$ as $h\to0$, $h\neq0$. We write $f'(x)=Df(x)=A$. A partial derivative takes this limit in one coordinate direction; the Jacobian matrix has those partials as its columns. The Jacobian determinant of a square map is $\det f'(x)$. A map is $C^1$ when its derivative is continuous; $C^k$ and smooth mean successive continuous differentiability through order $k$ and every finite order, respectively. One-variable derivatives at an interval endpoint are one-sided when specified. A local maximum or minimum compares the value with values in a neighborhood in the domain.

Riemann integrals, null sets and substitution

For $a<b$, a partition is $a=x_0<\cdots<x_q=b$. For bounded real $f$, put $m_j=\inf_{[x_{j-1},x_j]}f$, $M_j=\sup_{[x_{j-1},x_j]}f$. The lower and upper Darboux sums are $\sum_jm_j(x_j-x_{j-1})$ and $\sum_jM_j(x_j-x_{j-1})$. The lower integral is the supremum of lower sums; the upper integral is the infimum of upper sums. Equality defines Riemann integrability and the integral. The notation $\mathscr R(S)$ means the Riemann-integrable functions on $S$. A refinement adds partition points. A zero-length interval has integral zero; reversal changes the sign. Complex and finite-vector integrals are defined componentwise, with their norm inequalities proved in the local companions.

A rectangle in $\mathbb R^n$ is a Cartesian product of intervals. Its volume is the product of their lengths. A rectangle partition is a product of one-dimensional partitions; a subrectangle uses consecutive partition points. Darboux sums multiply each subrectangle's infimum or supremum by its volume. Their upper and lower integrals and the integrability definition are as above. The indicator $\chi_S$ is one on $S$ and zero elsewhere.

The outer measure $m^*(S)$ is the infimum of the sums of volumes of countable open-rectangle covers of $S$. The value $+\infty$ is allowed. A null set has outer measure zero: equivalently, for every $\varepsilon>0$ there is such a cover with total volume below $\varepsilon$, directly by the infimum definition. A bounded set is Jordan measurable when its indicator on a containing rectangle is Riemann integrable. Its Jordan volume is that integral. For a bounded function on a Jordan set, extend it by zero to a containing rectangle; integrability and the integral use this extension. The selected proofs establish independence of the containing rectangle and the other used properties.

For bounded real $f$ on a domain, its oscillation at $x$ is

Equivalently the infimum is the decreasing-radius limit by the monotone-limit proofs. Improper one-dimensional integrals take limits at excluded endpoints or infinity of the compact integrals. The multivariable absolute integrals, tails and product-majorant interchanges used by the stationary-phase lesson are defined and proved in P17–P18, rather than by an unspecified general integration theorem.

Exponentials and trigonometric notation

The source writes $L(x)=\int_1^x dt/t$ for $x>0$. Its proved inverse is $E$; these functions are $\log$ and the real exponential. For complex $z$, the exponential is the convergent series $\exp z=\sum_{j=0}^{\infty}z^j/j!$. Define $\cos z=(e^{iz}+e^{-iz})/2$ and $\sin z=(e^{iz}-e^{-iz})/(2i)$. The selected source proofs together with P13–P16 establish the series operations, derivatives, real period $2\pi$ and all root choices used here. No analytic-continuation theorem is assumed.

Definitions arranged by GPT-6 Astra (OpenAI), Ultra, 4 October 2026, from the selected freely accessible Lebl programme edition and its local completions. CC BY-SA 4.0. Human proof excerpts retain Jiří Lebl's authorship.