Interchange of limit and sum [closed]
Let $X:=(X(i))\in\ell^1(\mathbb{N})$ and $\{X_n\}$ be sequence in $\ell^1(\mathbb{N})$ such that(i) $X_n$ converges to $X$ in sup norm,(ii) $\sum_{i}X_n(i)$ is a Cauchy sequence.Then is it true that...
View ArticleFinding a function in the unit sphere of a functional subspace with a couple...
Preliminary:A={f $\in C(X); f(a)=0$} is a banach space with norm the following:$\Vert f\Vert=sup\vert f(x)-f(y)\vert; x,y \in X$( X is Hausdorff and compact space. 'a' is a point in X)$\tilde f (x,y)=...
View ArticleProve there are countable many discontinuities
Let I be a non-empty Interval and $f:I\rightarrow \mathbb{R}$ a monotone, non decreasing function. Show that $f$ has countable many Classification of discontinuities.My prove is nearly complete. But...
View ArticleOuter measure, $|A| = \lim_{t\to\infty} |A \cap (-t,t)|$ for all $A\subset...
I have already proved this result:If $A\subset R$ and $t>0$, then $|A| = |A\cap (-t,t)| + |A\cap(\mathbb{R}\setminus(-t,t))|$I would like to prove that $|A| = \lim_{t\to\infty} |A \cap (-t,t)|$ for...
View ArticleLebesgue integral of L^1 function is differentiable
Let $f\in L^1(\mathbb{R})$, and define the function$$F(x)=\int_a^xf(t)dt.$$I want to prove that $F$ is almost everywhere differentiable and that $F'(x)=f(x)$ where $F$ is differentiable.I am following...
View ArticleIf a sequence $a_n$ satisfies the following two properties, does...
Let $a_n$ be a positive, increasing sequence satisfying the following two properties:$S_k :=\displaystyle\sum_{n=1}^{\infty} \frac{1}{a_n^k}$ converges for all $k \in \mathbb{N}$.And...
View ArticleFinding $\min_{Q^TQ = I}(n Tr((Q^T \Lambda Q)^2) - Tr(Q^T \Lambda Q)^2)$
Let $n\in \mathbb{N}$, $c > 1$ and $T = T (n) = \lceil cn \rceil$. Let $p \in (0, 2]$. Givendiagonal $T \times T$ matrix $\Lambda$ with diagonal entries $\lambda_j =j^p$, $j = 1, \ldots, T$, let...
View ArticleProve the unbounded sequence $\left\{{a_n} \right\}$ has a subsequence...
Let $\left\{{a_n} \right\}$ be an unbounded sequence. Prove that thereexists a subsequence $\left\{{a_{p_n}} \right\}$ of $\left\{{a_n}\right\}$ such that $\left\{ \frac{1}{a_{p_n}} \right\}$ converges...
View ArticleIs my proof rigorous enough? Spivak, Prove:$\lim_{x \to a}f(x)=\lim_{h \to...
I'm aware that the answers to this question already exist on this site I would just like to know if my proof is rigorous enough (or incorrect).Prove that $\lim_{x \to a}f(x)=\lim_{h \to 0}f(a+h)$....
View ArticleExample for proper inclusion in Lorentz space
The details about Lorentz space are given in Wikipedia page. It is easy to prove that $L^{p,q}$ is contained in $L^{p,r}$ whenever $q<r$. I'm trying to construct an example which shows that the...
View ArticleProve, without using derivatives, that xsin(1/x) is not Lipschitz in (0,1]...
I know that it is not Lipschitz by using derivatives, but I can't prove it without.
View ArticleHow to show that the increment operation for natural numbers is well defined?
Let $a,b$ be natural numbers and $++$ be an increment operation. Based on Peano axioms alone, how to show that if $a=b$, then $a++ = b++$?Do note that the book I am reading (Real Analysis by Terrence...
View ArticleHow are these definitions of the limit superior and limit inferior equivalent?
I have come across these three definitions of the limit superior (or upper limit) and the limit inferior (or lower limit) of a sequence of real numbers and I wonder how to establish the equivalence of...
View ArticleReference Request: Hausdorff–Young inequality for the inverse Fourier seires
Let $ \hat f : \mathbb Z^d \to \mathbb C $ denote a function in $\ell^p(\mathbb Z^d)$ where $p \in [1,2]$.Let $f : \mathbb T^d = (\mathbb R / 2\pi \mathbb Z)^d \to \mathbb C$ denote the inverse Fourier...
View ArticleIs there a Log-Sobolev inequality for Lebesgue measure on $[0,1]$ on compact...
After searching on the internet for long enough, I would like to pose the question here. I hope there is no duplicate (if there is please let me know)Is it true that, there is a universal constant...
View ArticleHow to evaluate the integral $\int_0^{\infty} \frac{\sin(k \ln(t))}{\sqrt{t}}...
I am trying to evaluate the integral$$\int_{0}^{\infty}\frac{\sin\left(k\ln\left(t\right)\right)}{\sqrt{t}}\,\left\{\frac{1}{t}\right\} \, dt,$$ where $\left\{\cdots\right\}$ denotes fractional part...
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$z = xy$ intersects with $y = 2x^2$ on a curve. Points on curve are connected...
A surface in $\mathbb{R}^3$ with equation $z = xy$ intersects a surface with equation $y = 2x^2$ along some curve. Each point of the arc of this curve contained in the area $0 < z < 1$ is...
View ArticleIntegral $\int_x\Big|\frac{f(x)-f(0)}{x}\Big|^2\,dx$
How is there a functional inequality that allows me to bound $$\int_x\Big|\frac{f(x)-f(0)}{x}\Big|^2\,dx$$ in terms of a Sobolev norm of $f$, say?If $f$ is equal to its Taylor series at $0$, we have...
View Articleprove $\mathcal{L}^n(A(E))=|\det A|\mathcal{L}^n(E) $
suppose I have a linear map $A:\mathbb R^n \rightarrow \mathbb R^n$I want to prove $\mathcal{L}^n(A(E))=|\det A|\mathcal{L}^n(E) $$\forall E\subset \mathbb R^n$ where $\mathcal{L}^n$ is Lebesgue...
View ArticleUse $\,\varepsilon-\delta\,$ definition of limits to show that...
We need to show that for every $\varepsilon >0$ there exists a $\delta >0$ such that$$\left\lvert\frac{x^3+y^3}{x^2+y^2}-\frac{7}{5}\right\rvert<\varepsilon...
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