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In a Cauchy sequence, any two terms with large enough indices are as near to each other as we wish. A metric space is complete when each Cauchy sequence in it converges, and the real numbers form a complete space. The node Cauchy sequence gives the precise definition.

The second derivative of a function is the derivative of its derivative. It measures how the slope changes, and so how the graph bends. The node second derivative gives the precise definition.

A contour line of a function $f$ of two real variables is a curve in the plane along which $f$ has a constant value $c$. It is a level set $\{(x, y) : f(x, y) = c\}$ of $f$, drawn in the $(x, y)$-plane. A contour map draws the contour lines for several equally spaced values of $c$. It shows the shape of the graph of $f$ in the plane.

An alternating series is a series whose terms alternate in sign. Such a series converges when the absolute values of its terms decrease to zero. The node alternating series gives the precise definition.

A homeomorphism is a continuous bijection between topological spaces whose inverse is also continuous. Homeomorphic spaces are the same from the point of view of topology.

A totally disconnected space is a topological space whose only connected subsets are the empty set and the single points.

The Cauchy product of two series is the series whose $n$-th term is the sum of the products $a_k b_{n-k}$. It is the discrete convolution of the two sequences of terms. The node Cauchy product gives the precise definition.

A Hausdorff space is a topological space in which any two distinct points have disjoint open neighbourhoods. It is the most used separation axiom, and it makes limits of sequences unique.

A compact space is a topological space in which each open cover has a finite subcover. Closed bounded intervals of the real line are the basic examples.

A topological space is connected when it is not the union of two disjoint nonempty open sets. A connected space is in one piece.

A real sequence tends to infinity when its terms eventually exceed each given bound. Such a sequence diverges, and we say that its limit is $+\infty$. The node sequence tending to infinity gives the precise definition.

A continuously differentiable function is a differentiable function whose derivative is continuous. The classes $C^k$ ask for $k$ derivatives, the last one continuous. The node continuously differentiable function gives the precise definition.

A rearrangement of a series is the series of the same terms taken in a new order, given by a bijection of the indices. Rearrangements keep the sum of an absolutely convergent series. The node rearrangement of a series gives the precise definition.

The Riemann integral of a function on a closed bounded interval is the limit of its Riemann sums as the partitions get finer. For a nonnegative function, it is the area under the graph. The node Riemann integral gives the precise definition.

A convergent sequence is a sequence that has a limit. Convergent sequences are bounded, and their limits obey the algebra of limits. The node convergent sequence gives the precise definition.

A topological space is path-connected when any two of its points can be joined by a continuous path. Path-connected spaces are connected, but the converse is false.

The Borel σ-algebra of a topological space is the smallest σ-algebra that contains all open sets. Its elements are the Borel sets. On the real line it is the usual family of events for real random variables. The node Borel sigma-algebra gives the precise definition.

A series of functions is an infinite sum of functions defined on a common set. Its convergence is the convergence of its partial sums, pointwise or uniformly. The node series of functions gives the precise definition.

A sequence of functions is a sequence $(f_n)$ of functions $f_n \colon S \to \mathbb{R}$ on a common set $S$. For each fixed $x \in S$, the values $f_n(x)$ form a sequence of real numbers. Such sequences can converge in several ways, mainly pointwise or uniformly.

A series converges absolutely when the series of the absolute values of its terms converges. Absolutely convergent series converge, and their sums do not depend on the order of the terms. The node absolute convergence gives the precise definition.

A separable space is a topological space with a countable dense subset.

A block diagonal matrix is a square block matrix whose diagonal blocks are square and whose other blocks are all zero.

A lower triangular matrix is a square matrix whose entries above the main diagonal are all zero.

The Gram matrix of a finite family of vectors in an inner product space is the square matrix of all their pairwise inner products.

A linear subspace of a vector space is a subset that contains the zero vector and is closed under addition and scalar multiplication. It is a vector space with the same operations.

A positive semidefinite matrix is a Hermitian matrix $A$ with $x^* A x \ge 0$ for every vector $x$.

A nondegenerate bilinear form is a bilinear form for which no nonzero vector gives zero with every vector.

An elementary matrix is a matrix that we get from the identity matrix by one elementary row operation. Multiplying a matrix on the left by an elementary matrix performs that row operation.

A projection is a linear map from a vector space to itself that is equal to its own square. It projects the space onto its image along its kernel.

The column space of a matrix is the span of its columns. It is the image of the linear map of the matrix.

The minimal polynomial of an endomorphism of a finite-dimensional space is the monic polynomial of least degree that annihilates it.

A diagonal matrix is a square matrix whose entries outside the main diagonal are all zero.

A homogeneous system of linear equations is a system in which every constant term is zero. Such a system always has the zero solution, and its solution set is a vector subspace.

A nilpotent endomorphism is a linear map from a space to itself that becomes zero after finitely many compositions with itself.

A scalar matrix is a diagonal matrix whose diagonal entries are all equal, that is, a scalar multiple of the identity matrix.

The row space of a matrix is the span of its rows.

A nilpotent matrix is a square matrix $N$ such that $N^k = 0$ for some positive integer $k$.

The dual space of a vector space is the vector space of all linear forms on it.

A Jordan block is a square matrix with one scalar $\lambda$ on the diagonal, $1$ on the diagonal just above it, and $0$ elsewhere.

A linear isometry is a linear map between normed spaces that keeps the norm of every vector.