# 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.