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A Banach space is a normed vector space that is complete for the metric of its norm. Banach spaces are the main setting of functional analysis.

A circle integral is the contour integral of a function along a circle $|z - c| = R$, traveled once counterclockwise, see the definition.

An estimator is consistent when it converges to the true value of the parameter as the sample size grows. The node consistent estimator gives the precise definition.

Mathematical induction is a method to prove that a property $P(n)$ holds for every natural number $n$. A proof by induction has two steps. The base case proves $P(0)$. The induction step proves, for an arbitrary natural number $n$, that $P(n)$ implies $P(n + 1)$. Then $P(n)$ holds for every natural number $n$.

A proof by contrapositive proves an implication $p \to q$ by a direct proof of $\neg q \to \neg p$. The method is correct because $p \to q$ is logically equivalent to its contrapositive.

The arctangent is the inverse of the tangent function restricted to $(-\pi/2, \pi/2)$. It is defined on all of $\mathbb{R}$ and it is bounded. The node arctangent gives the precise definition.

The Frobenius automorphism of a finite field of characteristic $p$ is the map $x \mapsto x^p$. It generates the group of all automorphisms of the field. The definition of the Frobenius automorphism gives the details.

The arccosine is the inverse of the cosine function restricted to $[0, \pi]$. It is defined on $[-1, 1]$. The node arccosine gives the precise definition.

The joint distribution of several random variables on one probability space is the distribution of the vector that they form. It gives the probability of each event that involves several of the variables together. The node joint probability distribution gives the precise definition.

A division ring is a nonzero ring in which every nonzero element has a multiplicative inverse. It is like a field, but the multiplication need not be commutative. The definition of a division ring gives the conditions.

A proof by infinite descent shows that a statement has no solution in positive integers. It assumes a solution, and from it constructs a strictly smaller solution in positive integers. This gives an infinite decreasing sequence of positive integers, which is impossible.

A chi-squared test is a hypothesis test whose test statistic has approximately a chi-squared distribution under the null hypothesis. The main example is Pearson's chi-squared test for counts in categories. For observed counts $O_1, \dots, O_k$ and expected counts $E_1, \dots, E_k$ under $H_0$, Pearson's statistic is $$\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}.$$ One rejects $H_0$ when $\chi^2$ is large.

The hyperbolic sine is half the difference of $e^x$ and $e^{-x}$. It is an odd, strictly increasing function on $\mathbb{R}$. The node hyperbolic sine gives the precise definition.

A Student's t-test is a hypothesis test whose test statistic has Student's t-distribution under the null hypothesis. It mostly tests hypotheses about the means of normal populations with unknown variance. In the one-sample t-test, $X_1, \dots, X_n$ are i.i.d. $\mathcal{N}(\mu, \sigma^2)$ with unknown $\sigma$, and $H_0 \colon \mu = \mu_0$. The statistic is $$t = \frac{\bar{X}_n - \mu_0}{S_n / \sqrt{n}},$$ with the sample mean $\bar{X}_n$ and the sample standard deviation $S_n$. Under $H_0$, $t \sim t_{n - 1}$.

A proof by contradiction proves a proposition $p$ in two steps. It assumes $\neg p$, and it gets a contradiction from this assumption, that is, a statement $q$ together with $\neg q$. Then $\neg p$ is false, so $p$ is true by the double negation law.

The Maclaurin series of a smooth function is its Taylor series at the point $0$. It is a power series in $x$. The node Maclaurin series gives the precise definition.

A diffeomorphism is a bijection between open sets, or between manifolds, such that both the map and its inverse are differentiable. Diffeomorphisms are the changes of coordinates of calculus, and they preserve all differentiable structure.

The norm induced by an inner product is the square root of the inner product of a vector with itself.

A marginal distribution is the distribution of one variable, or of some of the variables, of a random vector, without regard to the other variables. The node marginal distribution gives the precise definition.

A linear recurrence with constant coefficients makes each term a fixed linear combination of the previous terms. Its solutions have closed forms in terms of the roots of its characteristic polynomial. The node linear recurrence with constant coefficients gives the precise definition.

A self-adjoint operator on a finite-dimensional inner product space is a linear map from the space to itself that equals its adjoint.

The sample variance of a sample measures the spread of its values around the sample mean. With the divisor $n - 1$, it is an unbiased estimator of the variance of the population. The node sample variance gives the precise definition.

A direct proof of an implication $p \to q$ assumes $p$ and then gets $q$ by a chain of valid steps. The steps use axioms, definitions and theorems that are already proved.

An estimator is a rule that computes a guess of an unknown parameter from the data. It is a statistic that one uses for this purpose. The node estimator gives the precise definition.

The prime subfield of a field is its smallest subfield. It is $\mathbb{Q}$ or $\mathbb{F}_p$, up to isomorphism, by the characteristic. The definition of the prime subfield gives the details.

A regular curve is a continuously differentiable parametrised curve whose tangent vector is never zero. At each point a regular curve has a well-defined tangent line, and it can be reparametrised by arc length.

A proof by cases splits the situation into a finite number of cases that cover all possibilities. Then it proves the statement in each case. The basic form is this rule: from $p \lor q$, $p \to r$ and $q \to r$, conclude $r$.

The hyperbolic tangent is the quotient of the hyperbolic sine by the hyperbolic cosine. It is a bounded, odd, strictly increasing function with range $(-1, 1)$. The node hyperbolic tangent gives the precise definition.

The multivariate normal distribution is the extension of the normal distribution to random vectors in $\mathbb{R}^k$. A mean vector and a covariance matrix determine it. The node multivariate normal distribution gives the precise definition.

The Taylor series of a smooth function at a point is the power series built from all the derivatives of the function at that point. For analytic functions, it converges to the function near the point. The node Taylor series gives the precise definition.

A finite field is a field with a finite number of elements. For each prime power $q$ there is exactly one finite field of order $q$, up to isomorphism, and there are no other finite fields. The definition of a finite field states it.

A one-sided limit is the limit of a function as the argument approaches a point from the left only or from the right only. A function has a limit at a point exactly when both one-sided limits exist and agree. The node one-sided limit gives the precise definition.

A countably infinite set is a set that is equinumerous with the set of natural numbers.

A strictly monotone function is a function that is either strictly increasing or strictly decreasing. Such a function is injective. The node strictly monotone function gives the precise definition.

A function is surjective when each element of the codomain is a value of the function.

The limit of a function at infinity is the value that the function approaches as its argument grows without bound. It describes horizontal asymptotes of graphs. The node limit at infinity gives the precise definition.

A subset of a metric space is totally bounded when, for each size, it can be covered by finitely many pieces of that size. Compact sets are exactly the totally bounded sets that are complete.

The discrete uniform distribution on a finite set gives the same probability to each element of the set. The node discrete uniform distribution gives the precise definition.

The Bernoulli distribution with parameter $p$ is the distribution of a random variable with two values. The value $1$ has probability $p$ and the value $0$ has probability $1 - p$. It is the distribution of one Bernoulli trial. The node Bernoulli distribution gives the precise definition.

A real random variable has a density when there is a fixed nonnegative function with this property. The probability of each set of values is the integral of the function over that set. The node random variable with a density gives the precise definition.