Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [analysis]

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

Mathematical analysis is the rigorous version of calculus. In fact, it investigates the theorems in calculus with enough care and deals with them more deeply, trying to generalize the ideas in calculus. You can consider a more specific tag instead: , , , , , , etc. For data analysis, use .

42884 questions
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Is a set without limit points necessarily closed?

According to the definition on Rudin' Principles of Mathematicial Analysis, closed set is defined as: $E$ is closed if every limit point of $E$ is a point of $E$. Then I have a question: if a set has no limit point, is it necessarily closed? I…
Brain Zhang
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Essential Supremum vs. Uniform norm

I just went to check something about the $||\cdot||_\infty$ norm and realized that it can perhaps refer to two quite different things. I'm coming at this from an Analysis class so I am use to having $||f||_\infty = ess sup |f(t)|$. But, according to…
Fractal20
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Convergence of finite differences to zero and polynomials

Assume that $f:\mathbb R \rightarrow \mathbb R$ is continuous and $h\in \mathbb R$. Let $\Delta_h^n f(x)$ be a finite difference of $f$ of order $n$, i.e $$ \Delta_h^1 f(x)=f(x+h)-f(x), $$ $$ \Delta_h^2f(x)=\Delta_h^1f(x+h)-\Delta_h^1…
A.B
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Covering number of the standard simplex

The covering number of the standard simplex is the minimal number of $k$-dimensional balls of radius $\epsilon$ that suffices for covering the probability simplex $\{x \in \mathbb{R}^{k} : x_1 + \dots + x_k = 1, x_i \ge 0, i=1, \dots, k \}$. A…
Anonymous
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Notation: What's meant by $C^{\infty}_{0}(\mathbb{R}^{+})$?

In Chapter 0 of Iwaniec's Spectral Methods of Automorphic Forms Iwaneic uses the notation $C^{\infty}_{0}(\mathbb{R}^{+})$ without definition. I assume that it's the set of infinitely differentiable functions from $\mathbb{R} \to \mathbb{R}$ with…
Jonah Sinick
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Why is volume of a high-D ball concentrated near its surface?

I came across the following sentence while reading a book on applied math: Volume of a high dimensional unit ball is concentrated near its surface and is also concentrated at its equator. This is from book's introduction, and I believe the…
Alex X
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Is power set of a power set of a set equal to the power set of the same set?

I have to decide whether this statement is true, I think it is not. Since the power set of a set with cardinality $n$, will have $2^n$ subsets, however the power set of this set will include the subsets themselves and subsets of the subsets.
user197848
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Terence Tao Exercise 5.4.3: Integer part of $x$ proof.

I am reading Terence Tao: Analysis 1. As you may be aware, certain objects are introduced bit by bit, so if i am not 'allowed' to use something yet, please understand. Show that for every real number $x$ there is exactly one integer $N$ such…
user214138
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How to know if a term is divisible by 10

I have some difficulties to solve this easy problem, could someone help me? Is $4^{1000}-6^{500}$ divisible by $10$?
Stefan Hanssen
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Show $e^{D}(f(x)) = f(x+1)$ where $D$ is the derivative operator

I would appreciate help showing $e^{D}(f(x)) = f(x+1)$ Where $D$ is the linear operator $D: \mathbb{C}[x] \rightarrow \mathbb{C}[x]$ where (in the context where this statement arose) $x \in \mathbb{N}$; $f(x) \mapsto \frac{d}{dx} f(x)$ By the…
user12802
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Continuous $f$, $\sup_{x,y\in R^m}\|f(x+y)-f(x)-f(y)\|<\infty$, then exist only one real matrix $A$, such that $f(x)-Ax$ is bounded.

Problem: f is a continuous function, $f: \mathbb{R}^m\to \mathbb{R}^n$, $\sup_{x,y\in R^m}\|f(x+y)-f(x)-f(y)\|<\infty$, there exist only one real matrix $A$, such that $f(x)-Ax$ is bounded. My thoughts 1: $\sup_{x,y\in…
Rogan
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Uncountable set with exactly one limit point

Is there any uncountable subset of $\mathbb{C}$ with exactly one limit point?
user90533
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Why the members of $\mathbb R$ will be certain subsets of $\mathbb Q$?

$\mathbb{R}$ is real numbers set, $\mathbb{Q}$ denotes rational numbers set. This is quoted from Rudin's mathematical analysis book page 17 about Dedekind' s construction. Why the members of $\mathbb{R}$ will be certain subsets of…
HyperGroups
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To construct a set with a limit point.

I learned how to construct a Cantor Set, and I am asked to do the following. "Construct a bounded set with exactly 3 limit points." Since the Cantor set contains infinitely many points, I don't think something like it will not work. But this is the…
hyg17
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Showing that $\mathbb{Q}$ is not complete

Show that there is no least upper bound for $A=\{x: x^2<2\}$ in $\mathbb{Q}$. Suppose $\alpha \in \mathbb{Q}$ is the least upper bound of $A$. Then either $\alpha^2 < 2$ or $\alpha^2 > 2$. I know the proof of the former case and why we can't have…
Xena
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