nLab modulus of convergence

Idea

A modulus of convergence is a function that tells how quickly a convergent sequence or net converges inside of a suitable Cauchy space, such as a metric space or uniform space.

Definition

The precise definition varies with the context.

In the real numbers, a modulus of convergence for a sequence or net $(x_i)_i$ is a function $\alpha$ from the positive numbers to the natural numbers (for sequences) or the directed set (for nets) such that almost all terms are within $\epsilon$ of one another. Explicitly:

$$\forall \epsilon,\; \forall i, j \geq \alpha(\epsilon),\; |x_i - x_j| \lt \epsilon .$$

In a metric space, a modulus of convergence for a sequence or net $(x_i)_i$ is a function $\alpha$ from the positive numbers to the natural numbers (for sequences) or the directed set (for nets) that satisfies the same condition, now relative to the metric $d$ on that space. Explicitly:

$$\forall \epsilon,\; \forall i, j \geq \alpha(\epsilon),\; d(x_i,x_j) \lt \epsilon .$$

In a gauge space, a modulus of convergence for a sequence or net $(x_i)_i$ is a function $\alpha$ from the positive numbers to the natural numbers (for sequences) or the directed set (for nets) that satisfies this condition for each gauging distance separately. Explicitly:

$$\forall d,\; \forall \epsilon,\; \forall i, j \geq \alpha(\epsilon),\; d(x_i,x_j) \lt \epsilon .$$

In a rational or real premetric space, a modulus of convergence for a sequence or net $(x_i)_i$ is a function $\alpha$ from the positive numbers to the natural numbers (for sequences) or the directed set (for nets) that satisfies the same condition, now relative to the premetric. Explicitly:

$$\forall \epsilon,\; \forall i, j \geq \alpha(\epsilon),\; x_i \sim_\epsilon x_j .$$

In a uniform space or preuniform space, a modulus of convergence for a sequence or net $(x_i)_i$ is a function $\alpha$ from the set of entourages to the natural numbers (for sequences) or the directed set (for nets) if an analogous condition is satisfied for each entourage $U$. Explicitly:

$$\forall U,\; \forall i, j \geq \alpha(U),\; x_i \approx_U x_j .$$

The most general space for which a modulus of convergence can be defined is a set $X$ with an auxiliary set $T$ and a ternary relation $R(x, y, t)$ for $x, y \in X$ and $t \in T$. In such a space, a modulus of convergence for a sequence or net $(x_i)_i$ is a function $\alpha$ from the auxiliary set to the natural numbers (for sequences) or the directed set (for nets) that satisfies the same condition for each element of the auxiliary set, now relative to the ternary relation. Explicitly:

$$\forall t,\; \forall i, j \geq \alpha(t),\; R(x_i, x_j, t) .$$

Related concepts

• sequence, net
• Cauchy sequence, Cauchy net
• limit of a sequence, limit of a net
• metric space, gauge space, uniform space
• Cauchy structure
• Cauchy approximation
• HoTT book real numbers

References

• Wikipedia, Modulus of convergence

• Auke Booij, Analysis in Univalent Type Theory (2020) [etheses:10411, pdf, pdf]

• Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)