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
1
vote
1 answer

Upper bound dependent on $n$

Could someone suggest an $n$-dependent but $x$-independent upper bound for $x^n(1-x^n)$ (used to show uniform convergence) on $[0,1]$? Thanks.
analysisnewbie
  • 13
1
vote
1 answer

Convergence of continuous linear maps

Let $E$ and $F$ be normed vector spaces and let $\lambda_{n}$ be a sequence in $L(E,F)$, the set of continuous linear maps from $E\rightarrow F$. Assume $F$ is complete. Let $v\in E$ and suppose that $\lambda_{n}(v)$ converges to a point in $F$.…
Mael
  • 741
1
vote
2 answers

Analysis convergence definition

When proving convergence the part where $|a_n-a|<\varepsilon$, is it necessary to use a strict inequality? Could I use $\leq$ instead? Thanks.
leonard
  • 13
1
vote
3 answers

convergence of a series in real analysis proof

If $\sum_{n=1}^{\infty}a_n$ converges, prove that $\sum_{n=1}^{\infty} \dfrac{n+1}{n}a_n$ converges. It's probably pretty trivial, but I have been staring at it for a while and cannot make any headway. Any help would be greatly appreciated
rebecca
  • 13
1
vote
2 answers

Closure regarding arithmetic mean

Say I have n integers bigger than zero with $n > 3$. I need to choose them in a way that leads to the arithmetic mean of any three of these integers to be one of the integers as well. I figured that for the arithmetic mean of any three integers to…
Bk1ng
  • 45
1
vote
1 answer

applying fubini's theorem

Let $I = [0 , 1]$; let $Q = I \times I$. Define $f: Q \to \mathbb{R}$ by letting $f(x , y) = 1 /q$ if y is rational and $X = p/q$, where p and q are positive integers with no common factor; let $f(x, y) = 0$ otherwise. need help with c)
user115722
  • 65
1
vote
1 answer

Trying to understand the construction of the $L^p$-spaces (factor spaces)

Let $(\Omega,\mathcal{A},\mu)$ be a measurabale space and consider the indicator function $1_B$ for $B\in\mathcal{A}$. Now I want to prove that $1_B\in L_{\mu}^p$ for all $p\geq 1$. But I have really big problems to show that, because I do not come…
user34632
1
vote
3 answers

Why does$A\subset \mathbb{Z}$ have a maximum value if it bounded from above?

Why does it happen that, if $A\subset \mathbb{Z}$ is bounded from above, it has a maximum value? How can it be explained?
evinda
  • 7,823
1
vote
2 answers

Function whose discontinuity points are a prefixed $F_\sigma$ set in $\mathbb{R}$.

I have been reading Carothers' book on real analysis and I found the following question on page 130: If E is an $F_\sigma$ set in $\mathbb{R}$, is $E=D(f)$ for some $f:\mathbb{R}\rightarrow \mathbb{R}$ ? Here $D(f)$ denotes the set of…
Julio
  • 203
  • 1
  • 7
1
vote
2 answers

How prove this $f'(c)=2c(f(c)-f(0))$

let $f(x)$ is continuous on $[0,1/2]$, and derivative on $(0,1/2)$,such $$f'(1/2)=0$$ show that there exsit $c\in(0,1/2)$, such $$f'(c)=2c(f(c)-f(0))$$ My try: let $$F(x)=e^{-x^2}[f(x)-f(0)]\Longrightarrow F'(x)=e^{-x^2}[f'(x)-2x(f(x)-f(0))]$$ then…
user94270
1
vote
1 answer

Cauchy product of series

There is a lemma for a certain proof in my notes: If $\sum_n |a_n| < \infty$ and $\sum_n |b_n| < \infty$ then $\sum_{n=0}^\infty \sum_{k=0}^n|a_k||b_{n-k}| < \infty$ Denote $ A_N = \sum_{n=0}^N |a_n|$ and $B_N = \sum_{n=0}^N|b_n|$ then since $A_N$…
DHx
  • 35
1
vote
3 answers

Convergent of p-series

My lecturer went through this method for proving $\sum_{n=1}^\infty \dfrac{1}{n^p} $ converges if $p>1$ Since $1/n^p > 0$ it implies that the sequence of partial sums $S_N$ is increasing. So either $S_n$ is bounded above & converges or it diverges…
DHx
  • 35
1
vote
1 answer

Approximation of continuous function uniformly by linear combination of constant functions on intervals

Let $f:[a,b] \rightarrow \mathbb R$ be a continuous function. I wish to prove that for $\varepsilon >0$ there are a subintervals $P_1,...,P_n \subset [a,b]$ and constants $c_1,...,c_n$ such that for $h=c_1 \chi_{P_1}+...+c_n \chi_{P_n}$ is…
Richard
  • 4,432
1
vote
1 answer

how to compute Möbius transformation's determinant

$SL(2, R)$ acts on $H^2$ by Möbius transformations $$ g\cdot z=\frac{az+b}{cz+d}, \quad g=\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix} \in SL(2,R), \quad z\in H^2,$$ where $H^2=\{z\in C \mid \operatorname{Im} (z)>0\}$, i.e., the complex upper…
user63788
  • 206
1
vote
1 answer

continuity problem

Let $f:\Bbb R \to\Bbb R$ be defined by setting $f(x) = \sin x$ if $x$ is rational, and $f(x) = 0$ otherwise. At what points is $f$ continuous? is this true? Let $c$ be irrational. Then there exists a sequence of rational numbers $x_n$ such that…
user115722
  • 65
1 2 3
…
99 100