Articles of convergence

In Hilbert space: $x_n → x$ if and only if $x_n \to x$ weakly and $\Vert x_n \Vert → \Vert x \Vert$.

Assume that $H$ is a $\mathbb K$-Hilbert space, $(x_n)_{n \ge 1}$ a sequence in $H$ and $x ∈ H$. Show that $x_n → x$ if and only if $x_n \to x$ weakly and $\Vert x_n \Vert → \Vert x \Vert$. I’m trying to prove this statement. The $\Rightarrow$ is basically clear, since a strongly convergent […]

Number of digits in row of Pascal's triangle is $O\left(n^2\right)$

Let $a_n$ be the number of decimal digits in the $n$-th row of Pascal’s Triangle (so $a_0=1, a_1=2, a_2=3, a_3=4, a_4=5, a_5=8,\dots$). Prove that $\frac{a_n}{n^2}$ converges and find the limit. It’s very easy to see that it converges. Indeed, it’s well known that $\binom{2n}{n}\sim\frac{4^n}{\sqrt{\pi n}}$ as $n\to\infty$. Letting $b_n=\log_{10}\binom{2n}{n}$, clearly $a_n\leq(n-2)b_n+2$. Then \begin{align*}\lim_{n\to\infty}\frac{a_n}{n^2}&\leq\lim_{n\to\infty}\frac{\log_{10}\frac{4^n}{\sqrt{\pi n}}}{2n}\\&=\lim_{n\to\infty}\frac{4n\ln(2)-1}{4n\ln(10)}\\&=\frac{\ln(2)}{\ln(10)}\approx0.301\dots\end{align*} However, […]

Prove the infinite sum $\sum_{k=0}^{\infty}{\frac{(2k-1)!!}{(2k)!!}x^{2k}}=\frac{1}{\sqrt{1-x^2}}$

I have been trying to prove that $$\sum_{k=0}^{\infty}{\frac{(2k-1)!!}{(2k)!!}x^{2k}}=\frac{1}{\sqrt{1-x^2}}$$ I have done it by using the binomial formula, bit we can’t use the gamma function and putting $-\frac{1}{2}$ is kinda logical but not that clear. I have also tried to calculate the partial sum formula for the series but it gets complicated so I can’t get […]

Normed space where all absolutely convergent series converge is Banach

This question already has an answer here: If every absolutely convergent series is convergent then $X$ is Banach 2 answers

Mollifiers: Asymptotic Convergence vs. Mean Convergence

Problem Does asymptotic convergence imply mean convergence: $$\varphi\in\mathcal{L}_\text{loc}(\mathbb{R}_+):\quad\varphi(T)\stackrel{T\to\infty}{\to}\varphi_\infty\implies\frac{1}{T}\int_0^T\varphi(s)\mathrm{d}s\stackrel{T\to\infty}{\to}\varphi_\infty$$ Remark Three important classes fall under local integrability: $\mathcal{L}(\mathbb{R}_+),\mathcal{C}(\mathbb{R}_+),\mathcal{B}(\mathbb{R}_+)\subseteq\mathcal{L}_\text{loc}(\mathbb{R}_+)$

Weak and strong convergence

I have the sequence $(v_n)\subset H^1_0(0,1)$ such that $v_n\rightharpoonup v $ (weakly) in $H^1_0(0,1)$ and $v_n\rightarrow v$ in $L^2(0,1)$ and $v_n\rightarrow v$ in $C^0(0,1)$ My question is why $$\int_0^1 v_n(x) (v_n(x)-v(x)) dx\rightarrow 0$$ I say that $\int_0^1 v_n (v_n-v) dx=\int_0^1 (v_n-v+v)(v_n-v) dx= \int_0^1 (v_n-v)^2 dx +\int_0^1 v(v_n-v) dx=$ $ ||v_n-v||^2_{L^2(0,1)} +\int_0^1 v(v_n-v) dx$ By the […]

Convergence, Integrals, and Limits question

Let $f: [0,\infty)\to \Bbb R$ be a positive,decreasing monotonic function. Prove the following statement for every a>0 providing the integral on the right side converges. First I managed to prove that the series on the left size converges using the integral test, but now I’m having a hard time proving that the equality above is […]

A sequentially compact subset of $\Bbb R^n$ is closed and bounded

Let $U$ be a subset of $\mathbb{R}^n$, and suppose that $U$ is not bounded. Construct a sequence of points $\{a_1, a_2, \ldots \}$ such that no subsequence converges to a point in $U$, then prove this claim about your sequence. Let $U$ be a subset of $\mathbb{R}^n$ such that $U$ is not closed. Construct a […]

Proof: A convergent Sequence is bounded

A part of the proof says that if $n\le N$, then, the sequence $x_n \le \max\{|x_1|,|x_2|,….,|x_{N-1}|\}$. I’m not capturing the intuition of the above. Even more perplexing is, if it the case where the sequence is monotonic decreasing, then for every $n\le N$, it is obvious that $x_n$ will not be $\le \max\{|x_1|,|x_2|,….,|x_{N-1}|\}$. I greatly […]

Convergence of a sequence in two different $L_p$ spaces

If I have a sequence $\{f_n\}_{n\in \mathbb{N}} \subset L_p\cap L_q$ that is Cauchy under both norms, I am wondering if $f_n \to f$ in $L_p$ implies that $f_n\to f$ in $L_q$. I have been working on this but I can’t seem to figure out a definite proof. Thank you! PS: $1\leq p<q <\infty$