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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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Decreasingly directed families

We have the Lemma If $\alpha_i \uparrow \alpha$, $\beta_i \uparrow \beta$, then $\alpha_i\beta_i \uparrow \alpha\beta$. And I have to show The analogue of this Lemma for decreasing directed families is false. I thought that repeating the proof…
user2820579
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Sequence problem, find root

the equation $x^3-5x+1=0$ has a root in $(0,1)$. Using a proper sequence for which $$|a(n+1)-a(n)|\le c|(a(n)-a(n-1)|$$ with $0
Plom
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the set $\mathbb N$ of natural numbers is an infinite set consisting of finite numbers

Tao Analysis For instance, it is possible to have an infinite set consisting of finite numbers (the set $\mathbb N$ of natural numbers is one such example), and it is also possible to have a finite set consisting of infinite objects (consider for…
Andrew Li
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Proof global minimum of $x^2+y^4-y^2$

Proof global minimum of $f(x,y)=x^2+y^4-y^2$ is $\frac{1}{2}$ where $f:$ $\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$. Is my proof attempt correct? Define $g(y):=f(0,y)$. Differentiating $g$ after $y$ we find $4y^{3}-2y$. By setting…
Iwan5050
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In Terence Tao "Analysis", the definition of the addition of natural numbers refers to a previous definition of $m+n$, but I can't find it.

Has anyone read about Terence Tao's Analysis? Definiton 2.2.1 Addition of natural numbers ... " From our discussion of recursion in the previous section, we see that we have defined $n + m$ for every natural number $n$. Here we are specializing the…
Andrew Li
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Applications of Stone-Weierstrass Theorem, excercise

Suppose that $f\in\mathscr{C}_{\mathbb{R}}\left([0,1]\right)$, show that \begin{equation} \lim\limits_{n \to \infty}\frac{\int_0^1 x^{n}f(x)dx}{\int_0^1 x^{n}dx}=f(1) \end{equation} My idea is to use the Stone-Weierstrass Theorem but I'm a little…
Dagoberto Mares
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Induction Using Multi-Index

Does anyone know how to use induction in the context of multi-indices? I know the induction is done on the multi-index length, the main problem is how to split a multi-index of length $n+1$ into one of length $n$ and another of length $1$. For…
PtF
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Need pre-requisites for Machine Learning by Kevin P Murphy

Hi and thanks for taking the time to read my question. I have a goal to read Kevin P Murphy's Machine Learning, A probabilistic Perspective. I have only read half way through an introductory real analysis textbook (Steven L Ray's Analysis with an…
Cai
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Why is the set of all integers not open?

A point $p$ is an interior point of a set $E$ if there exists a neighborhood $N_r(p)$ such that $N_r(p)\subseteq E$, and a set is open if all of its points are interior points. Now my question is that since $r\in \mathbb{R^+}$ then why can't we…
Melanie
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How can one prove that if $f$ is linear with a closed kernel, then $f$ is continuous

Let $(V,\|.\|)$ be a normed vector space, $ f:V\to \mathbb{R} $ be linear and $ \ker(f) $ closed. Then $f$ is continuous. How does one prove this? My idea was to distinguish between two cases. Because $ \ker(f) $ is closed all convergent sequences $…
hallo007
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Interaction of two values

I am a mathematically-challenged guy struggling (or to say better - having no clue at all) about a problem. I have two values (let's call them value ONE and TWO). The first can go from 55 to 190. The second can be anything i want. By making these…
user86345
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Is the set of linear functions from $[0,1]$ to $\mathbb{R}$ equicontinuous?

By the set of linear functions I mean the functions of the form $$f(x)=\alpha x,$$ for $\alpha\in\mathbb{R}$. We clearly have for any $x,y\in[0,1]$ $$|f(x)-f(y)|=|\alpha||x-y|.$$ So for any $\epsilon>0$, $\delta=\frac{\epsilon}{|\alpha|}$ gives…
Bryce Smith
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Commutative composition of functions

Let $f$ and $g$ be two functions, both of them from $\Bbb R$ to $\Bbb R$, defined by the formulas: $ f: x \mapsto x\sqrt{x^2+4} $ and $g: x \mapsto x^3 + 3x \\ $ I saw, in an exercise, that $f \circ g = g \circ f$, but even if I can compute it, I…
Jérémie Delage
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Divergence and curl both zero. What can we say about behavior at infinity?

All right, so my Professor claims that it is impossible to find a vector field that has both div and curl zero and yet vanishes at infinity (that is, becomes the zero vector). I don't know if it is correct. I decide to trust my professor and check…
Math Monk
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Absolutely continuous function Sobolev space

Today we had a $L^1((0,A);\mathbb{R})$-function $G: (0,A)\rightarrow \mathbb{R}$ with $G(a)=\int_0^ag(s)ds$ for all $a\in (0,A)$ which is absolutely continuous, $A>0$. Why is $G\in W^{1,1}(0,A;\mathbb{R})$ then? They only said proof with…
Uhmm
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