Arzela-Ascoli Theorem: Is only pointwise boundedness required?
In Royden's text the Arzela-Ascoli Theorem states:Let X be a compact metric space and $f_n$ a uniformly bounded, equicontinuous sequence of real valued functions on X. Then $f_n$ has a subsequence that...
View ArticleIs there a notation for normed spaces $(X, \| \cdot \|_X)$ and $(Y, \| \cdot...
Suppose that we are dealing with two abstract nomed spaces $(X, \| \cdot \|_X )$ and $(Y, \| \cdot \|_Y)$ such that $ X = Y $ (that is, every element of $X$ belongs to $Y$ and every element of $Y$...
View Articleexistence of a limit implies that a function can be extended to a harmonic...
TheoremIf $u: \mathbb{D'} = \mathbb{D} \setminus \{0\} \to \mathbb{R}$ is harmonic and bounded, then $u$ extends to a function harmonic in $\mathbb{D}$.In the next proof $\Pi^+$ is the upper...
View ArticleHow does measure theory deal with higher cardinalities?
The second part of the definition of a sigma-algebra is that countable unions of measurable sets are measurable. The second property of a measure is that the measure of countable unions of measurable...
View ArticleWhy are Lebesgue integrals defined as a supremum and not as a limit?
We may approximate a bounded nonnegative function $f$ by simple functions $\phi$. Then the standard way of defining the Lebesgue integral of $f$ is$$ \int f d\mu := \sup \Bigg\{ \int \phi : \phi \leq...
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Proving piecewise function is k-differentiable function
For this problem,I have a question about the solution,Do you please know how they go from the $f^{n}(x)$ to $f^{n + 1}(x)$ step (I assume they differentiate $f^{n}(x)$)? I think they made a typo there...
View ArticleStep in proof of Heine-Borel Theorem
I am continuing my work on Lindstrom's real analysis book Spaces (those who have answered my past questions know this by now), and I am working on exercise 2.3.10 which asks for a proof of a special...
View ArticleIs this approach to prove Dirichlet princple correct?
I want to prove that for open region $\Omega\subset \mathbb{R}^d$ with certain boundary regularity and continuous function $f$, the Dirichlet problem$$\Delta u = 0, \quad u|_{\partial \Omega} = f$$has...
View ArticleExamples of uncountable sets with zero Lebesgue measure
I would like examples of uncountable subsets of $\mathbb{R}$ that have zero Lebesgue measure and are not obtained by the Cantor set.Thanks.
View Article$\frac{\prod_{j=1}^{∞}\frac{2j}{2j-1}}{\lim\limits_{n \to...
How to solve $$\frac{\prod_{j=1}^{∞}\frac{2j}{2j-1}}{\lim\limits_{n \to ∞}\Gamma(\frac{1}{n})} = ?$$Where $\Gamma(x)$ is the Gamma function.
View ArticleIf $f(x)=x^2 \sin(1/x^2)$ for $x\neq 0$, $f(0)=0$ then $\int_0^1 |f'(x)| dx =...
I am trying to prove if $f(x)=x^2 \sin(1/x^2)$ for $x\neq 0$, $f(0)=0$ then $\int_0^1 |f'(x)| dx = \infty$so $f'\notin L^1$This came up in Rudin's Real and Complex Analysis Section 7.16 (a).I tried...
View ArticlePointwise limit of continuous functions whose graph is in a given closed set
Let $C\subseteq\mathbb R^2$ be a closed set with the property that for every $x\in\mathbb R$, there exists at least one $y\in\mathbb R$ such that $(x,y)\in C$.Does there exist a function $f:\mathbb...
View ArticleFunction $\frac{x}{1+x}g(a-x) + \frac{1}{1+x}g(-x)$ has a unique maximizer...
Let $g:(-L,\infty) \to \mathbb{R}$ be a strictly concave, increasing, and differentiable function with $L > 0$ such that $g(x)\to -\infty$ as $x \to -L$. Also, let the map $f$ be defined on $[0,L)$...
View ArticleVariations of Dawson's function
I am studying Dawson's function : $\displaystyle F : x \mapsto e^{-x^2}\int_0^x e^{t^2} dt$.I would like to prove that $F$ attains a maximum at a certain value $x_0 \in (0,1)$, and is increasing over...
View ArticleHow to use local approximation spaces to build a global space via the...
I'm working to understand how the partition of unity method is used to build a global PUM space from local approximation spaces. Can someone please explain the mechanics of gluing the local spaces...
View Articler^n converges for r=1. what behaviour it will show for r
The sequence (r^ n) converges for;help me solvewhat i tried to do with this is i took the first case ascase1: 0<=r<1 we take 0 as positive term series since r lies between 0 and 1 so the series...
View ArticleShow that $\mathring{\mathbb Z}=\emptyset$
I want to find interior of $\mathbb{Z}$ in $\mathbb{R}$. Here is the definition of $\mathring{A}$:$$\mathring{A}=\{x\in \mathbb{R}\mid \exists r>0:(x-r,x+r)\subseteq A\}$$My proof:Suppose on...
View ArticleProperties of the reciprocal of a series expansion
Suppose that a given a polynomial function $f(x) = \sum_{j=0}^N (-1)^j a_j x^j$ of degree $N$ analytic in $x=0$ with $1>a_1>a2>\ldots> a_N>0$, and such that the series expansion of...
View ArticleUniform convergence of sequence of partial sums.Help please
Show that each sequence of partial sums and its derivative converges uniformly on their respective intervals.a) $$ S_n(x)=\sum_{k=1}^n \frac{1}{(1 + nx)^2}, x \in (0,\infty)$$$$ S_n'(x)=\sum_{k=1}^n...
View ArticleContinuity on the torus
Let $\mathbb{T}^d$ be the $d$-dimensional torus, for all $r >0,\eta_r(x):=e^{-r|x|^2},x \in \mathbb{T}^d,$ we denote by $\mathscr{F}^{-1}\eta_r$ the inverse Fourier transform defined by...
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