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Let \(\displaystyle{\left(H,\langle{,\rangle}\right)}\) be a Hilbert space. If \(\displaystyle{U:H\to H}\) is a \(\displaystyle{\mathbb{C}}\) - linear and bounded operator such that \(\displaystyle{||U||\leq 1}\), then prove that \(\displaystyle{\left(\forall\,h\in H\right)\,\,\left(U(h)=h\iff U^{\s...

i. Suppose that \(\displaystyle{d(x,A)=\inf\,\left\{d(x,y)\geq 0\,,y\in A\right\}=0}\). For every \(\displaystyle{n\in\mathbb{N}}\), there exists \(\displaystyle{y_n\in A}\) such that \(\displaystyle{d(x,y_n)<\dfrac{1}{n}}\). The sequence \(\displaystyle{\left(y_n\right)_{n\in\mathbb{N}}\subseteq A...

Hi Tredy. Firstly, if \(\displaystyle{x\,,y\in G}\), then \(\displaystyle{\begin{aligned}x\star y+\dfrac{1}{2}&=x+y+2\,x\,y+\dfrac{1}{2}\\&= \dfrac{2\,x+1}{2}+\dfrac{2\,y\,(2\,x+1)}{2}\\&=\dfrac{(2\,y+1)\,(2\,x+1)}{2}\\&\neq 0\end{aligned}}\) so, \(\displaystyle{x\star y\neq -\dfrac{...

Here is another idea (similar to the previous one) . We define \(\displaystyle{F:R/I\to R/J}\) by \(\displaystyle{F(x+I)=\phi(x)+J}\) If \(\displaystyle{x+I=y+I\in R/I}\), then \(\displaystyle{x-y\in I}\), so \(\displaystyle{\phi(x-y)\in \phi(I)=J\implies \phi(x)-\phi(y)\in J\implies \phi(x)+J=\phi(...

Let \(\displaystyle{\left(f_{m}\right)_{m\in\mathbb{Z}}}\) be a complex sequence such that \(\displaystyle{\sum_{m\in\mathbb{Z}}|f_m|<\infty}\). Consider the continuous function \(\displaystyle{f:\left(-\pi,\pi\right]\to \mathbb{C}}\) defined by \(\displaystyle{f(t)=\sum_{m\in\mathbb{Z}}f_m\,e^{i\,m...

Examine if there exists a \(\displaystyle{1-1}\) function \(\displaystyle{f:\mathbb{N}\to \mathbb{N}}\)

such that \(\displaystyle{\sum_{n=1}^{\infty}\dfrac{f(n)}{n^2}<\infty}\)

Let \(\displaystyle{\left(R,+,\cdot\right)}\) be an associative ring with unity \(\displaystyle{1=1_{R}}\) and \(\displaystyle{I\,,J}\) double ideals of \(\displaystyle{\left(R,+,\cdot\right)}\) . If there exists \(\displaystyle{\phi\in Aut(R)}\) such that \(\displaystyle{\phi(I)=J}\), then prove th...

Solve the problem \(\displaystyle{u_{tt}(x,t)-u_{xx}(x,t)=0\,,(x,t)\in\left(0,\pi\right)\times \left(0,+\infty\right)}\), where \(\displaystyle{u(x,0)=x\,,0\leq x\leq \pi\,\,,u_{t}(x,0)=\sin\,x\,,0\leq x\leq \pi\,\,,u(0,t)=u(\pi,t)=0\,,t\geq 0\,\,\,,u\in C^{\infty}}\) Find \(\displaystyle{M=\max\,\l...

Hi Riemann. Thank you for your answer.

Examine if the following is true.

If \(\displaystyle{p}\) is a prime number and\(\displaystyle{\left(G,\cdot\right)}\) is a group

such that \(\displaystyle{o(G)=p^3}\) and \(\displaystyle{x^p=1\,,\forall\,x\in G}\) then

\(\displaystyle{\left(G,\cdot\right)}\) is abelian.