*Part 5 of a six-part series of adaptations from Terence Tao’s book “Structure and Randomness".*

## Prerequisites

A rough idea of

*big O notation*would also be helpful but is not necessary. A brief introduction: we say that $f(x) = O(g(x))$ (as $x \to \infty$) to mean that the "growth" of $f(x)$ is "bounded above" by $g(x)$; more formally, there exist positive real numbers $M$ and $x_0$ such that $\lvert f(x)\rvert \leq M\lvert g(x)\rvert$ for all $x > x_0$.

*Note that $\lvert x\rvert$ denotes the absolute value of $x$, not its cardinality as indicated in Part 4.*

## [1.3] Soft analysis, hard analysis, and the finite convergence principle

*Analysis*(something like an advanced calculus) is often differentiated into "hard analysis" ("quantitative", "finitary") and "soft analysis" ("qualitative", "infinitary").

*Discrete math*,

*computer science*, and

*analytic number theory*normally uses hard analysis while

*operator algebra*,

*abstract harmonic analysis*, and

*ergodic theory*tend to rely on soft analysis. The field of

*partial differential equations*uses techniques from both.

Convenient notation (e.g. $O(\:)$) from qualitative analysis can conceal gritty details from quantitative and argue efficiently from the big picture, at the cost of a precise description. Conversely, quantitative analysis can be seen as a more precise and detailed refinement of qualitative analysis. The intuitions, methods and results in hard analysis often have analogues in soft analysis and vice versa, despite their contrasting language. Tao argues this technique transfer can benefit both disciplines. Table 5 features a rough "dictionary" between the notational languages of soft and hard analysis. Kudos to Tao for such an illuminating comparison!

Soft analysis | Hard analysis |

$x$ is finite | $x$ is bounded (e.g. $x = O(1)$) |

$x$ vanishes | $x$ is small (e.g. $\lvert x\rvert \leq \varepsilon$) |

$x$ is infinite | $x$ is large (e.g. $\lvert x\rvert \geq N$) |

$x_n \to 0$ | Quantitative decay bound (e.g. $x_n = O(n^{-c})$) |

$x_n$ is convergent | $x_n$ is metastable* |