Definition:Convergence
Convergent Sequence
Topological Space
Let $T = \struct {S, \tau}$ be a topological space.
Let $A \subseteq S$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be an infinite sequence in $A$.
Then $\sequence {x_n}$ converges to the limit $\alpha \in S$ if and only if:
- $\forall U \in \tau: \alpha \in U \implies \paren {\exists N \in \R_{>0}: \forall n \in \N: n > N \implies x_n \in U}$
Metric Space
Let $M = \struct {A, d}$ be a metric space or a pseudometric space.
Let $\sequence {x_k}$ be a sequence in $A$.
$\sequence {x_k}$ converges to the limit $l \in A$ if and only if:
- $\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \map d {x_n, l} < \epsilon$
For other equivalent definitions of a convergent sequence in a Metric Space see: Definition:Convergent Sequence in Metric Space
Normed Division Ring
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\sequence {x_n} $ be a sequence in $R$.
The sequence $\sequence {x_n}$ converges to the limit $x \in R$ in the norm $\norm {\, \cdot \,}$ if and only if:
- $\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - x} < \epsilon$
Normed Vector Space
Let $\struct {X, \norm {\,\cdot \,} }$ be a normed vector space.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $X$.
Let $L \in X$.
The sequence $\sequence {x_n}_{n \mathop \in \N}$ converges to the limit $L \in X$ if and only if:
- $\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: \forall n \in \N: n > N \implies \norm {x_n - L} < \epsilon$
Test Function Space
Let $\map \DD {\R^d}$ be the test function space with the compact support $K \subseteq \R^d$.
Let $\sequence {\phi_n}_{n \mathop \in \N}$ be a sequence in $\map \DD {\R^d}$.
Let $\phi \in \map \DD {\R^d}$ be a test function.
Let $D^k := \dfrac {\partial^{k_1 + k_2 + \ldots + k_d}} {\partial x_1^{k_1} \partial x_2^{k_2} \ldots \partial x_d^{k_d} }$ be a partial differential operator with the multiindex $k = \tuple {k_1, k_2, \ldots, k_d}$.
Suppose:
- $\forall n \in \N : \forall x \in \R^d \setminus K : \map {\phi_n} x = 0$
Suppose $\sequence {\phi_n}_{n \mathop \in \N}$ converges uniformly to $\phi$.
Suppose that for every multiindex $k$ the sequence $\sequence {D^k \phi_n}_{n \mathop \in \N}$ converges uniformly to $D^k \phi$.
Then the sequence $\sequence {\phi_n}_{n \mathop \in \N}$ converges to $\phi$ in $\map \DD {\R^d}$.
This can be denoted:
- $\phi_n \stackrel \DD {\longrightarrow} \phi$
Schwartz Space
Let $\map \SS \R$ be the Schwartz space.
Let $\sequence {\phi_n}_{n \mathop \in \N}$ be a sequence in $\map \SS \R$.
Let $\phi \in \map \SS \R$ be a Schwartz test function.
Suppose:
- $\ds \forall l, m \in \N : \lim_{n \mathop \to \infty} \sup_{x \mathop \in \R} \size {x^l \map {\phi_n^{\paren m} } x} = 0$
where:
- $\phi^{\paren m}$ denotes the $m$th derivative of $\phi$
- $\sup$ denotes the supremum.
Then the sequence $\sequence {\phi_n}_{n \mathop \in \N}$ converges to $\mathbf 0$ in $\map \SS \R$.
This can be denoted:
- $\phi_n \stackrel \SS {\longrightarrow} \mathbf 0$
Real Numbers
Let $\sequence {x_k}$ be a sequence in $\R$.
The sequence $\sequence {x_k}$ converges to the limit $l \in \R$ if and only if:
- $\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \size {x_n - l} < \epsilon$
where $\size x$ denotes the absolute value of $x$.
Rational Numbers
Let $\sequence {x_k}$ be a sequence in $\Q$.
$\sequence {x_k}$ converges to the limit $l \in \R$ if and only if:
- $\forall \epsilon \in \R_{>0}: \exists N \in \R_{>0}: n > N \implies \size {x_n - l} < \epsilon$
where $\size x$ is the absolute value of $x$.
Complex Numbers
Let $\sequence {z_k}$ be a sequence in $\C$.
$\sequence {z_k}$ converges to the limit $c \in \C$ if and only if:
- $\forall \epsilon \in \R_{>0}: \exists N \in \R: n > N \implies \cmod {z_n - c} < \epsilon$
where $\cmod z$ denotes the modulus of $z$.
Convergent Series
Let $\struct {S, \circ, \tau}$ be a topological semigroup.
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be a series in $S$.
This series is said to be convergent if and only if its sequence of partial sums $\sequence {s_N}$ converges in the topological space $\struct {S, \tau}$.
If $s_N \to s$ as $N \to \infty$, the series converges to the sum $s$, and one writes $\ds \sum_{n \mathop = 1}^\infty a_n = s$.
Convergent Series in a Normed Vector Space (Definition 1)
Let $V$ be a normed vector space.
Let $d$ be the induced metric on $V$.
Let $\ds S := \sum_{n \mathop = 1}^\infty a_n$ be a series in $V$.
$S$ is convergent if and only if its sequence $\sequence {s_N}$ of partial sums converges in the metric space $\struct {V, d}$.
Convergent Series in a Normed Vector Space (Definition 2)
Let $\struct {V, \norm {\, \cdot \,} }$ be a normed vector space.
Let $\ds S := \sum_{n \mathop = 1}^\infty a_n$ be a series in $V$.
$S$ is convergent if and only if its sequence $\sequence {s_N}$ of partial sums converges in the normed vector space $\struct {V, \norm {\, \cdot \,} }$.
Convergent Series in a Number Field
Let $S$ be one of the standard number fields $\Q, \R, \C$.
Let $\ds \sum_{n \mathop = 1}^\infty a_n$ be a series in $S$.
Let $\sequence {s_N}$ be the sequence of partial sums of $\ds \sum_{n \mathop = 1}^\infty a_n$.
It follows that $\sequence {s_N}$ can be treated as a sequence in the metric space $S$.
If $s_N \to s$ as $N \to \infty$, the series converges to the sum $s$, and one writes $\ds \sum_{n \mathop = 1}^\infty a_n = s$.
A series is said to be convergent if and only if it converges to some $s$.
Convergent Mapping
Metric Space
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $c$ be a limit point of $M_1$.
Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$ defined everywhere on $A_1$ except possibly at $c$.
Let $\map f x$ tend to the limit $L$ as $x$ tends to $c$.
Then $f$ converges to the limit $L$ as $x$ tends to $c$.
Real Function
As the real number line $\R$ under the usual (Euclidean) metric forms a metric space, the definition also holds for real functions:
Let $f: \R \to \R$ be a real function defined everywhere on $A_1$ except possibly at $c$.
Let $\map f x$ tend to the limit $L$ as $x$ tends to $c$.
Then $f$ converges to the limit $L$ as $x$ tends to $c$.
Complex Function
As the complex plane $\C$ under the usual (Euclidean) metric forms a metric space, the definition also holds for complex functions:
Let $f: \C \to \C$ be a complex function defined everywhere on $\C$ except possibly at $c$.
Let $f \left({z}\right)$ tend to the limit $L$ as $z$ tends to $c$.
Then $f$ converges to the limit $L$ as $z$ tends to $c$.
Convergent Filter
Let $\struct {S, \tau}$ be a topological space.
Let $\FF$ be a filter on $S$.
Then $\FF$ converges to a point $x \in S$ if and only if:
- $\forall N_x \subseteq S: N_x \in \FF$
where $N_x$ is a neighborhood of $x$.
That is, a filter converges to a point $x$ if and only if every neighborhood of $x$ is an element of that filter.
If there is a point $x \in S$ such that $\FF$ converges to $x$, then $\FF$ is convergent.
Convergent Filter Basis
Let $\struct {S, \tau}$ be a topological space.
Let $\BB$ be a filter basis of a filter $\FF$ on $S$.
Then $\BB$ converges to a point $x \in S$ if and only if:
- $\forall N_x \subseteq S: \exists B \in \BB: B \subseteq N_x$
where $N_x$ is a neighborhood of $x$.
That is, a filter basis is convergent to a point $x$ if every neighborhood of $x$ contains some set of that filter basis.
Also see
- Results about convergence can be found here.