Evaluating a combinatorial sum
Find an expression/upper bound for $$\sum_{r\le X}\frac{1}{m^r}{{k+r-1}\choose{r}}$$where $m,k$ are fixed.Looking at the first few terms, we see that this is the same as...
View ArticleCounterexample for familiy of equicontinuous and pointwise bounded functions...
I aware of the theorem such that"Let $(f_n)$ equicontinuous and pointwise bounded sequence of real-valued functions on a compact metric space then $(f_n)$ is uniformly bounded."What if I change the...
View ArticleProve that $x_n \rightarrow a \Rightarrow (x_n)^{\frac{1}{\alpha}}...
One of my friends asked me this question while doing a proof for convergence and this has confused me quite a bit. While this can be proved when $\frac{1}{\alpha} \in \mathbb{Z}$ easily by algebra of...
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Family of separable Hilbert spaces over locally compact form a continuous...
Let $\{H_{x}\}_{x\in G^{0}}$ be a family of separable Hilbert spaces and $G^{0}$ be a locally compact second countable topological space. Let $\mathbb{B}_{x}$ be the orthonormal basis of $H_{x}$.If we...
View ArticleProof of Mean value theorem in $R^{n}$ using line segment trick
For reference, my book is Mutlidimensional analysis. The page is 56. The proof of Mean value theorem in my book goes as follows. Firstly, consider a lemma(I'll state without proof).Consider an open set...
View ArticleLusin's Theorem for finite measure spaces
Lusin's Theorem. Let $f$ be a real-valued measurable function on $E$. Then for each $\varepsilon>0$, there is a continuous function $g$ on $\mathbb{R}$ and a closed set $F$ contained in $E$ for...
View ArticleIs it true that the $\sqrt{3}$ can be written as an infinite fraction? How...
Is it true that the $\sqrt{3}$ can be written as an infinite fraction? How can we prove that?I was doing Real Analysis, Algebra and Calculus the other day, and I've read in a book that any irrational...
View ArticleIs $k_d(x)=\inf(\{d(x,b):b\in B\})$ uniformly continuous? [duplicate]
Let $d:\mathbb{R^2}\to\mathbb{R}$ be a metric on $\mathbb{R}.$ Let $B$ be a non-empty subset of $\mathbb{R}.$ Define $k_d:\mathbb{R}\to\mathbb{R}$ given by $k_d(x)=\inf(\{d(x,b):b\in B\}.$ Is $k_d$...
View ArticleProve that $\Bbb R$ satisfies Nested Interval Property (NIP) implies it...
Prove that $\Bbb R$ satisfies Nested Interval Property (NIP) implies it satisfies Axiom of Completeness (AoC) as well.I tried solving this problem hereby:We consider $S\subset \mathbb R$ such that...
View ArticleSequences with tails in $l^1$
Suppose that $a_n \geq a_{n+1} > 0$ and $\sum_{n \geq 0}a_n < \infty$. If the tails of the sequence$t_N = \sum_{n \geq N}a_n$are also in $l^1$, can anything be said about the rate at which $a_n...
View ArticleWhy is the sequence $(t^n)$ an example of Cauchy but not convergent in...
Is it not correct that the sequence converges to the zero function on $[0,1]$ which is in $C[0,1]$ in the $L^1$ sense, that is: the area between the $n$th monomial and the $0$ function goes to $0$ as...
View ArticleEstablishing the relationship among different double integrals.
Consider$f : [0, 1] \to [0, \infty)$ and$h : [0, 1] \to \mathbb{R}$,i.e., $h$ can be both positive and negative.Define $A = [0,1] \times [0,1]$.Consider the following integrals:\begin{align*}I_{1}&...
View ArticleRule for convolution of trigonometric functions?
Consider the following functions: for $x \in [-1/2,1/2] \subset \mathbb{R}$, $f_{0}(x) = 1$, $f_{2k-1}(x) = \sqrt{2}\sin((2k-1)\pi x)$ and $f_{2k}(x) = \sqrt{2}\cos(2k\pi x)$, for $k \in...
View ArticleIs the given function convergent?
Given a function $$y=\sqrt{log(x)+\sqrt{log(x)+\ldots}}$$ then what is the value of $\frac{dy}{dx}$.My approach was to assume the inner radical as $y$ and then by equation manipulation we get...
View ArticleProof of the Equivalence of Sequential Continuity and Continuity for a...
Theorem 1.3.2A linear form $u$ on $C_c^\infty(X)$ is a distribution if and only if $\lim_{j \to \infty} \langle u, \phi_j \rangle = 0$ for every sequence $\phi_j$ which converges to zero in...
View ArticleA weighted averaging inequality for continuous integrable functions [closed]
Let $a\in (0,1)$ and $f$ be a continuous and integrable function on $[0,\infty)$. Is it true that$$\left|\int_{0}^{1} (1-t)^{-a}{f(x t)}dt \right|\geq C |f(x)|$$for all $x>0$, for some constant $C$...
View ArticleFix distribution $u$, find a distribution $\tilde u$ satisfying $x\cdot\tilde...
Prove that, for every distribution $u\in\mathcal D'(\mathbb R)$,there exists a distribution $\tilde u\in \mathcal D'(\mathbb R)$ satisfying $x\cdot\tilde u=u$.It seems that it can be solved by Fourier...
View Articletheorem of regularity of Lebesgue measurable set
I was reading a proof regarding the condition for Lesbesgue measurable set. Specifically it is the Theorem 2.24 and the proof of the theorem here:...
View ArticleSolving $\left\lfloor \sin \left( x\right) \right\rfloor -\left\lfloor \sin...
I've created this equation $\left\lfloor \sin \left( x\right) \right\rfloor -\left\lfloor \sin \left( \left\lfloor x\right\rfloor \right) \right\rfloor =1$ for real numbers $x$. I've found a...
View ArticleFinding the function with power series $ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}...
I want to evaluate the power series $$ f(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n^2}. $$I am getting that $f(x)$ is equal to some integral of $\dfrac{\ln(x+1)}x$, which is not elementary...
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