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Landscape of 21st Century Mathematics

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Abstract

This chapter presents a collection of theorems in mathematical analysis, proved in the twenty-first century, which are at the same time great and easy to understand. The chapter is written for undergraduate and graduate students interested in mathematical analysis, as well as for mathematicians working in other areas of mathematics, who would like to learn about recent achievements in mathematical analysis without going into technical details.

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Notes

  1. 1.

    A complex number \(z\in {\mathbb C}\) is a critical point of a polynomial P if P′(z) = 0. It is non-degenerate if P″(z) ≠ 0.

  2. 2.

    For any sets \(C_1 \subset \mathbb R\) and \(C_2 \subset \mathbb R\), their arithmetic difference is the set C 1 − C 2 = {x − y | x ∈ C 1, y ∈ C 2}.

  3. 3.

    A set S is called connected if it cannot be partitioned into two disjoint open sets. A set S is called locally connected if for every z ∈ S and every open set U containing z there is an open set V  containing z such that the intersection V ∩ S is connected and contained in U.

  4. 4.

    A probability space is a triple \((X, \mathcal {B}, \mu )\), where X is a set, \(\mathcal {B}\) is a σ-algebra over X (sets in \(\mathcal {B}\) are called measurable ), and \(\mu :\mathcal {B}\to [0,1]\) is a probability measure , that is, a measure such that μ(X) = 1 and μ(∅) = 0.

  5. 5.

    L p(X) denotes the set of functions \(g:X \to {\mathbb R}\) such that ∫X|g(x)|pdμ < ∞.

  6. 6.

    A polynomial \(Q(x) \in {\mathbb Z}[x]\) is called a divisor of \(P(x) \in {\mathbb Z}[x]\) if P(x) = Q(x)R(x) for some \(R(x)\in {\mathbb Z}[x]\). If \(P(x) \in {\mathbb Z}[x]\) cannot be written as P(x) = Q(x)R(x) for non-constant \(Q(x),R(x) \in {\mathbb Z}[x]\), we say that P(x) is irreducible. .

  7. 7.

    A monic polynomial is a polynomial with leading coefficient 1.

  8. 8.

    We say that a random variable X possesses finite moments of all orders if the expectation of |X|k is finite for all k > 0.

  9. 9.

    A linear form in n variables in an expression of the form L = c 1 x 1 + ⋯ + c n x n, where c 1, c 2, …, c n are some complex coefficients.

  10. 10.

    The integer part \(\left \lfloor {z}\right \rfloor \) of a real number z is the largest integer not exceeding z.

  11. 11.

    Here, \(\langle x,y\rangle =\sum _{i=1}^n x_i y_i\) denotes the inner product in \({\mathbb R}^n\).

  12. 12.

    A function \(f:{\mathbb R}\to {\mathbb R}\) is called nowhere monotone if there are no real numbers a < b such that f is monotone on (a, b).

  13. 13.

    A level set of a function \(u:{\mathbb R}^n \to {\mathbb R}\) is a set of the form

    $$\displaystyle \begin{aligned} L_c(f)=\{(x_1, \dots, x_n)\,|\,u(x_1, \dots, x_n)=c\} \end{aligned}$$

    for some constant \(c\in {\mathbb R}\).

  14. 14.

    A function \(f:{\mathbb R}\to {\mathbb R}\) is called a Schwartz function if there exist all derivatives f (k)(x) for all k = 1,  2,  3, … and for all \(x\in {\mathbb R}\), and, for every k and \(\gamma \in {\mathbb R}\), there is a constant C(k, γ) such that \(|x^\gamma f^{(k)}(x)| \leq C(k,\gamma ), \, \forall x\in {\mathbb R}\).

  15. 15.

    A function \(f:{\mathbb R} \to {\mathbb R}\) is called even if f(−x) = f(x) for all \(x\in {\mathbb R}\), and odd if f(−x) = −f(x) for all \(x\in {\mathbb R}\).

  16. 16.

    A function \(\phi : {\mathbb R}^n \to \overline {\mathbb R}\) is lower-semicontinuous if the set \(\{x\in {\mathbb R}^n\big |f(x)\leq c\}\) is closed for all \(c \in {\mathbb R}\).

  17. 17.

    A map \(B:{\mathbb R}^n \to {\mathbb R}^n\) is called linear if B(x + y) = B(x) + B(y) and B(αx) = αB(x) for all \(x,y \in {\mathbb R}^n\) and \(\alpha \in {\mathbb R}\), symmetric if 〈Bx, y〉 = 〈x, By〉 for all \(x,y \in {\mathbb R}^n\), and invertible if B(x) ≠ 0 whenever x ≠ 0.

  18. 18.

    A map \(T:l^2(\mathbb Z) \to l^2(\mathbb Z)\) is called invertible if for every \(y \in l^2(\mathbb Z)\) there exists a unique \(x \in l^2(\mathbb Z)\) such that T(x) = y.

  19. 19.

    See the discussion before Theorem 3.7 for the definition of Cantor sets.

  20. 20.

    See (1.22) for the definition of the Hausdorff measure. In fact, for integer n, it coincides with the Lebesgue measure, up to a constant factor.

  21. 21.

    Recall that a norm is a function \(B \to {\mathbb R}\) such that (i) ∥x∥B ≥ 0 for all x ∈ B, (ii) ∥x∥B = 0 if and only if x = 0, (iii) ∥αx∥B = |α|∥x∥B for every scalar α, and (iv) ∥x + y∥B ≤∥x∥B + ∥y∥B for all x, y ∈ B. A normed space B is called complete if every Cauchy sequence converges to some limit in B.

  22. 22.

    A Banach space B is called separable if it contains a sequence x 1, x 2, …, x n, … such that for any x ∈ B and any 𝜖 > 0 there exists an n such that ||x − x n||B < 𝜖.

  23. 23.

    Recall that a metric space X is called complete if every Cauchy sequence in X converges to some limit in X.

  24. 24.

    By the L p-space we mean, for concreteness, the space of functions \(f:[0,1]\to {\mathbb R}\) with norm \(||f||{ }_p=\left (\int _0^1|f(x)|{ }^p\mathrm {d}x\right )^{1/p}<\infty \).

  25. 25.

    Intuitively, \(\mathcal {H}_1\) is just “length” of an arc, see (1.22) with f(r) = r for the formal definition.

  26. 26.

    That is, \(\mathcal {B}\) is a family of subsets of some set X such that (i) \(X\setminus A \in \mathcal {B}\) for all \(A\in \mathcal {B}\), (ii) \(\emptyset \in \mathcal {B}\), and (iii) \(A \cup B \in \mathcal {B}\) for all \(A,B \in \mathcal {B}\).

  27. 27.

    Recall that a function \(f:{\mathbb R}^n \to {\mathbb R}^m\) is called K-Lipschitz if ||f(x) − f(y)||m ≤ K||x − y||n for all \(x,y\in {\mathbb R}^n\), where ||⋅||n is the Euclidean norm in \({\mathbb R}^n\).

  28. 28.

    As usual, H n is the n-dimensional Hausdorff measure, see (1.22) for the definition.

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    Bogdan Grechuk

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Grechuk, B. (2021). Analysis. In: Landscape of 21st Century Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-80627-9_3

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