Let E be the set of vectors in $R^5$ whose sum of absolute valued components...
Here's the problem formatted:Formatted ProblemWhat I've tried so far:Heine Borel tells us that it suffices to check if the set is closed and bounded.Bounded: This seems obvious to me. This set can be...
View ArticleShow that $\sum_{k=1}^{\infty} kx^{k-1}$ converges uniformly on $[a,b]$ for...
I wish to show that $\sum_{k=1}^{\infty} kx^{k-1}$ converges uniformly on $[a,b]$ for any $-1<a<b<1$.Clearly this series is the derivative of the geometric series , so my thought is to use the...
View ArticleClik here to view.
$M \subset \mathbb{R}^3$ is a rotation of set $\{ x^2 + z^2 = 4x-3, 1
$M \subset \mathbb{R}^3$ is a surface created by rotation of set$\{ x^2 + z^2 = 4x-3, \ 1<x<2, \ -1<z<1 \}$ by $\pi$ around axis OZ.Find $\int_M \max(x,y,z) d \lambda_2$.I know that $x^2 +...
View ArticleProving the Lp norm is a norm.
I want to prove the $L_p$ norm on continuous functions is in fact a norm. I have proven definitiveness and homogeneity but am struggling with the triangle inequality. I am using the fact that if the...
View ArticleIf $f \in C^2((a, b)) \cap C^1([a, b])$, then $f$ is convex on $[a, b]$ if...
I'm trying to solve one of the questions from An Introduction to Functional Analysis by James C. Robinson:Show that if $f : [a, b] \to \mathbb{R}$ is $C^2$ on $(a, b)$ and $f$ is $C^1$ on $[a, b]$,...
View ArticleWeierstrass function is a Hölder function
The Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. The function was firstly described in 1872. Its original definition is as...
View ArticleA continuous extension of an uniformly continuos function on an interval
Let $a, b \in \mathbb{R}$ such that $a<b$ and let $f:(a,b)\to\mathbb{R}$ be a continuous function. Prove that are equivalenti) There exists a continuous function $g:[a,b]\to\mathbb{R}$ such that...
View ArticleClik here to view.
Papa Rudin Theorem $7.15$.
There is the definition of $(D\mu)(x)$:There is the theorem:If $\mu$ is a Borel measure on $R^k$ and $\mu \ \bot \ m$, then $$(D\mu)(x) = \infty \ a.e. [\mu]. $$ (Denote this equality by $(1)$.)There...
View ArticleMinimum of $\frac{x^3}{x-6}$ for $x>6$ without using derivative? [closed]
Edit with the hope of saving this post from closure (25/April)When looking for the minimum of a function, calculus is the default choice (with good reason), but it is a relatively new idea in...
View ArticleLebesgue decomposition of increasing function on an unbounded interval?
The Lebesgue decomposition theorem for functions of bounded variation states: If $f : I \to \mathbb{R}$ is a right-continuous function of bounded variation on an interval $I$ then $f$ has a...
View ArticleHow to use Finite Decimal Continuity to Define Real Number Arithmetic?
Here is some context for my question: In "Vector Calculus, Linear Algebra and Differential Forms" by Hubbard & Hubbard, in Appendix A.1 on real number arithmetic they construct what we mean by real...
View ArticleA question on proving an inequality involving a sequence of real numbers
Let $a_n$ be a sequence of real numbers such that $1=a_1 \le a_2 \le a_3 \le \cdots \le a_n.$Additionally, we have that $a_{i+1}-a_i \le \sqrt{a_i},$ for all $1 \le i <n.$Then prove that...
View ArticleCurious about what my real analysis teacher means when she uses $1 \leq k < j...
In the solutions she provided following our first midterm, my real analysis professor used this this summation:$$\sum_{1\leq k<j \leq n,} a_kb_ka_jb_j$$ I have never seen this type of terminology...
View ArticleBound of derivatives of test function
Let $f$ be a smooth function of compact support on $\mathbb R$.Let$$a_n=\sup_{x\in\mathbb R}|f^{(n)}(x)|.$$How does the sequence $a_n$ grow as $n\to\infty$? Is it bounded by $c^n$ for some $c>0$?
View ArticleEvaluating a rational function integral in a quick way
In an recent test I was asked to evaluate the integral$$ \int_0^1 \frac{\sqrt[3]{x^2(1-x)}}{(1+x)^3} \text{d}x$$in 8 minutes, but I didn't have a clue what to do with it.After the test, I tried the...
View ArticleIs it possible to find a closed form for $x! (a-x)!$?
I wonder if there is a closed form for $x! (a-x)!$? or lets focus on $\Gamma(x) \Gamma(a-x)$ instead of the factorial.$$B(x,y)= \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$$so $\Gamma(x) \Gamma(a-x)=...
View ArticleAlternatives to Serge Lang Real and Functional Analysis
I am a master's student and I am taking an analysis course, one of the reference books is this one. I have a hard time reading the book because of its structure, so I wanted to know if anyone has an...
View ArticleUsing Mean Value Theorem to prove that $f$ continuous and $\lim_{x\to...
I was trying to prove the following statement:Given $f$ continuous on an interval around $x_0$, and that $\lim_{x\to x_0}{f^\prime(x)}=L$ for finite $L$, $f^\prime(x_0)$ exists and equals $L$.This...
View ArticleShow that for $x > 1$, the series $\sum \frac{2n}{1 + x ^{2n} }$ converges...
enter image description hereShow that for $x > 1$, the series $\sum \frac{2n}{1 + x ^{2n} }$ converges uniformly to the function $(x-1 )^{-1}$.
View ArticleA Generalisation of Holomorphness
A holomorphic function is a function for which there exists a derivation in $\Bbb{C}$. If you consider it as a real function $\Bbb{R}^2\to\Bbb{R}^2$, its jacobian is a scaled rotation matrix.In this...
View Article