Articles of real analysis

How to prove that $\int_0^b\Big(\int_0^xf(x,y)\;dy\Big)\;dx=\int_0^b\Big(\int_y^bf(x,y)\;dx\Big)\;dy$?

Problem. Let $f:[0,b]\times[0,b]\to\mathbb{R}$ be continuous. Prove that $$\int_0^b\left(\int_0^xf(x,y)\;dy\right)\;dx=\int_0^b\left(\int_y^bf(x,y)\;dx\right)\;dy.\tag{1}$$ My first thought was to use Fubini’s theorem: the left hand side of $(1)$ equals the double integral $$\iint_{D_1} f(x,y)\;dA\tag{2}$$ and the right-hand side equals $$\iint_{D_2} f(x,y)\;dA,\tag{3}$$ where $D_1=\{(x,y);\;0\leq x\leq b,\;0\leq y\leq x\}$ and $D_2=\{(x,y);\;0\leq y\leq b,\;y\leq x\leq b\}$. Since $D_1=D_2$, the integrals $(2)$ and $(3)$ are […]

example of maximal operator that is integrable

We know that there are no nonzero functions $f \in L^1(\mathbb R^n)$ such that $Mf \in L^1(\mathbb R^n)$, where $Mf$ is the Hardy Littlewood maximal function. Can we find a maximal operator that is integrable for nonzero functions? More precisely, I would really appreciate some help with the following: Let $ \phi \in C^\alpha (\mathbb […]

How to prove that $\sum_{n=1}^\infty{n^2a^{n-1}}=\frac{1+a}{(1-a)^3}$

I want to prove that $$\sum_{n=1}^\infty{n^2a^{n-1}}=\frac{1+a}{(1-a)^3}$$ I start off at the sum and try to work my way into the equation. I know that the sum is: $$σ_{n}=1+2^2+a+3^2a^2+4^2a^3+\dots+n^2a^{n-1} (1)\Leftrightarrow$$ $$aσ_{n}=a+2^2a^2+3^2a^3+\dots+n^2a^n (2)$$ If I subtract (2) from (1), I get: $$(1-a)σ_{n}=1+2^2a+3^2a^2+\dots+n^2a^{n-1}-a-2^2a^2-\dots-n^2a^n \Leftrightarrow$$ $$(1-a)σ_{n}=(n^2+(n-1)^2)a^{n-1}-n^2a^n \Leftrightarrow$$ $$σ_{n}=\frac{n^2-(n-1)^2-n^2a^n}{1-a} \Leftrightarrow$$ $$σ_{n}=-\frac{1+n^2a^n}{1-a}$$ and that is what I’ve got so far. How […]

Are bounded linear maps continuous?

Let $B(X, Y)$ be the set of bounded linear maps from $X$ to $Y$ (i.e. such that $\sup_{||x|| \leq 1} L(x) < \infty$). Is $L \in B(X, Y)$ continuous? What about if $X$ is a Banach space? What about if $Y$ is a Banach space? Thank you!

Surface area with cavalier's principle

The formula for the lateral surface area of an oblique cylinder as shown in the picture (linked because I can’t post images) is $$A=2\pi ra$$ or $$A =\frac{2\pi rh}{\sin(v)}$$ oblique cylinder This makes intuitive sense by Cavalieri’s principle, but how would one prove it explicitly by integration? Is there a parametrisation for the cylinder which […]

How to prove that $b^{x+y} = b^x b^y$ using this approach?

Fix $b>1$. If $m$, $n$, $p$, $q$ are integers, $n > 0$, $q > 0$, $r = m/n = p/q$, then I can prove that $(b^m)^{1/n} = (b^p)^{1/q}$. Hence it makes sense to define $b^r = (b^m)^{1/n}$. I can also show that $b^{r+s} = b^r b^s$ if $r$ and $s$ are rational. If $x$ is […]

Books for starting with analysis

I am interested in self-studying real analysis and I was wondering which textbook I should pick up. I have knowledge of all high school mathematics, I have read How to Prove It by Daniel J. Velleman (I did most of the excercises) and I have completed a computational calculus course which covered everything up to […]

Proving an asymptotic lower bound for the integral $\int_{0}^{\infty} \exp\left( – \frac{x^2}{2y^{2r}} – \frac{y^2}{2}\right) \frac{dy}{y^s}$

This is a follow up to the great answer posted to Let $ 0 < r < \infty, 0 < s < \infty$ , fix $x > 1$ and consider the integral $$ I_{1}(x) = \int_{0}^{\infty} \exp\left( – \frac{x^2}{2y^{2r}} – \frac{y^2}{2}\right) \frac{dy}{y^s}$$ Fix a constant $c^* = r^{\frac{1}{2r+2}} $ and let $x^* = x^{\frac{1}{1+r}}$. […]

Increasing sequence of step functions converging to $\chi_G$ where $G$ is open in $$

Proposition 1. Let $I=[a,b]$ be a compact interval of $\mathbb{R}$. Let $G$ be an open set of $I$. Then there exists an increasing sequence of step functions $\{s_n\}$ defined on $I$ such that $s_n\nearrow \chi_{G}$ almost everywhere on $I$. I was trying to prove it, so I first considered the case of $G$ being connected […]

$\sup\{g(y):y\in Y\}\leq \inf\{f(x):x\in X\}$

Let $X$ and $Y$ be two nonempty sets and let $h:X\times Y\rightarrow \mathbb{R}$ have a bounded range in $\mathbb{R}$.Let $f:X\rightarrow \mathbb{R}$ and $g:Y\rightarrow \mathbb{R}$ defined by $$f(x)=\sup\{h(x,y):y\in Y\}$$ and $$g(y)=\inf\{h(x,y):x\in X\}$$Then can we prove that $$\sup\{g(y):y\in Y\} \leq \inf\{f(x):x\in X\}?$$