Bernhard Riemann- "If only I had a theorems! Then I should find the proofs enough." 
Formula:
To be specific, we say that the Riemann integral of f equals s if the following condition holds: For all ε > 0, there exists δ such that for any tagged partition ${\displaystyle x_{0},\ldots ,x_{n}}$ and ${\displaystyle t_{0},\ldots ,t_{n-1}}$ whose mesh is less than δ, we have

${\displaystyle \left|\sum _{i=0}^{n-1}f(t_{i})(x_{i+1}-x_{i})-s\right|<\varepsilon .}$

 Georg Friedrich Bernhard Riemann (German: [ˈʀiːman]  17 September 1826 – 20 July 1866) was an influential German mathematician who made lasting and revolutionary contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis. His famous 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded, although it is his only paper in the field, as one of the most influential papers in analytic number theory. Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity.

 Agustus De Morgan-"Mathematicians care no more for logic than logicians for mathematics"
Formula:

• Hyperbolic sine:

${\displaystyle \sinh x={\frac {e^{x}-e^{-x}}{2}}={\frac {e^{2x}-1}{2e^{x}}}={\frac {1-e^{-2x}}{2e^{-x}}}}$


               Augustus De Morgan (/dɪ ˈmɔːrɡən/; 27 June 1806 – 18 March 1871) was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous.

De Morgan was a brilliant and witty writer, whether as a controversialist or as a correspondent. In his time there flourished two Sir William Hamiltons who have often been conflated. One was Sir William Hamilton, 9th Baronet (that is, his title was inherited), a Scotsman, professor of logic and metaphysics at the University of Edinburgh; the other was a knight (that is, won the title), an Irishman, professor at astronomy in the University of Dublin. The baronet contributed to logic, especially the doctrine of the quantification of the predicate; the knight, whose full name was William Rowan Hamilton, contributed to mathematics, especially geometric algebra, and first described the Quaternions. De Morgan was interested in the work of both, and corresponded with both; but the correspondence with the Scotsman ended in a public controversy, whereas that with the Irishman was marked by friendship and terminated only by death. In one of his letters to Rowan, De Morgan says,

Be it known unto you that I have discovered that you and the other Sir W. H. are reciprocal polars with respect to me (intellectually and morally, for the Scottish baronet is a polar bear, and you, I was going to say, are a polar gentleman). When I send a bit of investigation to Edinburgh, the W. H. of that ilk says I took it from him. When I send you one, you take it from me, generalize it at a glance, bestow it thus generalized upon society at large, and make me the second discoverer of a known theorem.

The correspondence of De Morgan with Hamilton the mathematician extended over twenty-four years; it contains discussions not only of mathematical matters, but also of subjects of general interest. It is marked by geniality on the part of Hamilton and by wit on the part of De Morgan.

Stefan Banach-"Mathematics is the most beautiful and most powerful creation of the human spirit"

Formula:Sequential Banach-Alaoglu theorem
  let X be a separable normed space and B the closed unit ball in X^∗. Since X is separable, let {x_n} be a countable dense subset. Then the following defines a metric for x, y ∈ B

 

 Stefan Banach ([ˈstɛfan ˈbanax]  30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the world's most important and influential 20th-century mathematicians. He was one of the founders of modern functional analysis, and an original member of the Lwów School of Mathematics. His major work was the 1932 book, Théorie des opérations linéaires (Theory of Linear Operations), the first monograph on the general theory of functional analysis.

Banach's dissertation, completed in 1920 and published in 1922, formally axiomatized the concept of a complete normed vector space and laid the foundations for the area of functional analysis. In this work Banach called such spaces "class E-spaces", but in his 1932 book, Théorie des opérations linéaires, he changed terminology and referred to them as "spaces of type B", which most likely contributed to the subsequent eponymous naming of these spaces after him.The theory of what came to be known as Banach spaces had antecedents in the work of the Hungarian mathematician Frigyes Riesz (published in 1916) and contemporaneous contributions from Hans Hahn and Norbert Wiener. For a brief period in fact, complete normed linear spaces were referred to as "Banach-Wiener" spaces in mathematical literature, based on terminology introduced by Wiener himself. However, because Wiener's work on the topic was limited, the established name became just Banach spaces.
Likewise, Banach's fixed point theorem, based on earlier methods developed by Charles Émile Picard, was included in his dissertation, and was later extended by his students (for example in the Banach–Schauder theorem) and other mathematicians (in particular Brouwer and Poincaré and Birkhoff). The theorem did not require linearity of the space, and applied to any Cauchy space (complete metric space).
The Hahn–Banach theorem, is one of the fundamental theorems of functional analysis.

• Banach–Tarski paradox
• Banach–Steinhaus theorem
• Banach–Alaoglu theorem 
• Banach–Stone theorem