“There are some things which cannot be learned quickly, and time, which is all we have, must be paid heavily for their acquiring. They are the very simplest things, and because it takes a man’s life to know them the little new that each man gets from life is very costly and the only heritage he has to leave.” - Ernest Hemingway (More…)
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Collected Lectures on Real Analysis
📝 MIT OpenCourseWare Lectures on Calculus - G. Strang 📝 Elementary Calculus: An Approach Using Infinitesimals - Professor H. Jerome Keisler 📝 An Introduction to Real Analysis - John K. Hunter (University of California at Davis) 📝 Introduction to Real Analysis - William F. Trench (Trinity University, Texas) 📝 Basic Analysis: Introduction to Real Analysis - Jiří Lebl 📝 Elementary Real Analysis - Thomson, Bruckner 📝 Lecture Notes in Real Analysis - Eric T. Sawyer (McMaster University) 📝 Real Analysis - C. McMullen 📝 Real Analysis for Graduate Students - Richard F. Bass 📝 Modern Real Analysis - William P. Ziemer (Indiana University) 📝 Mathematical Analysis Vol I - Elias Zakon 📝 Mathematical Analysis Vol II - Elias Zakon 📝 Advanced Calculus - Lynn Loomis, Schlomo Sternberg 📝 Analysis of Functions of a Single Variable - Lawerence Baggett 📝 The Calculus of Functions of Several Variables - Dan Sloughter 📝 A ProblemText in Advanced Calculus - John M. Erdman 📝 Calculus and Linear Algebra. Vol. 1 - Wilfred Kaplan, Donald J. Lewis 📝 Calculus and Linear Algebra. Vol. 2 - Wilfred Kaplan, Donald J. Lewis 📝 Introduction to Calculus I and II - J.H. Heinbockel 📝 Active Calculus - Matt Boelkins 📝 Supplements to the Exercises in Chapters 1-7 of Walter Rudin’s “Principles of Mathematical Analysis” - George M. Bergman 📝 Calculus Made Easy - Silvanus P. Thompson (1910) 📝 Elements of Differential and Integral Calculus - William Anthony Granville (1911) 📝 Precalculus - Carl Stitz, Jeff Zeager
Ebooks on Combinatorics
Metric $k$-center
General $k$-center problem statement: Let \((X, d)\) be a metric space where \(X\) is a set and \(d\) is a metric. A set \(V \subseteq X\) is provided together with a parameter \(k\). The goal is to find a subset \(C \subseteq V\) with \(|C| = k\) such that the maximum distance of a point in \(V\) to the closest point in \(C\) is minimized. The problem can be formally defined as follows: Input: a set $V \subseteq X$, and a parameter $k$. Output: a set $C \subseteq V$ of $k$ points. Goal: Minimize the cost $r^C(V) = \max_{v \in V} d(v, C)$ The k-Center Clustering problem can also be defined on a complete undirected graph $G = (V, E)$ as follows:
A lemma of J. L. Lions
This post explores J. L. Lions’ lemma about Banach spaces with compact injection, including applications to functional analysis. Lemma statement: Let $X$, $Y$, and $Z$ be three Banach spaces with norms $|| \cdot ||_X$, $|| \cdot ||_Y$, and $|| \cdot ||_Z$. Assume that $X \subset Y$ with compact injection and that $Y \subset Z$ with continuous injection. Prove that $$ \forall \varepsilon > 0, \exists C_\varepsilon > 0 \text{ satisfying } || u ||_Y \leq \varepsilon || u ||_X + C _{\varepsilon}|| u ||_Z,\quad \forall u \in X $$
Complex Hahn-Banach Theorem
Let $X$ be a complex vector space, $X_0$ one of its subspaces, $p: X \to \mathbb{R}_+$ such that $$ p(\lambda x) = |\lambda| p(x), \quad \forall \lambda \in \mathbb{C}, x \in X \text{ and } p(x + y) \leq p(x) + p(y), \quad \forall x, y \in X, $$ satisfying $|f(x)| \leq p(x)$, $\forall x \in X_0$, where $f: X_0 \to \mathbb{C}$ is linear. Under these conditions, there exists a linear functional $F: X \to \mathbb{C}$ such that $F|_{X_0} = f$ and