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Hi r9m. Here is another solution. Since \(\displaystyle{||U||\leq 1}\), we get \(\displaystyle{||U(x)||\leq ||x||\,,\forall\,x\in H}\). Let \(\displaystyle{h\in H}\). Suppose that \(\displaystyle{U(h)=h}\). Then, \(\displaystyle{\begin{aligned}||U^{\star}(h)-h||^2&=\langle{U^{\star}(h)-h,U^{\sta...

For \(\displaystyle{p\in\left[1,+\infty\right)}\), consider \(\displaystyle{E_{p}:=\left\{f\in L^{p}([0,+\infty))\,\,,\int_{0}^{\infty}f(x)\,\mathrm{d}x=0\right\}}\). (Lebesgue measure) i. Prove that \(\displaystyle{E_{p}}\) is a linear subspace of \(\displaystyle{L^{p}([0,+\infty))}\). ii. Prove th...

Let \(\displaystyle{\left(X,||\cdot||\right)}\) be a normed space (real or complex) and \(\displaystyle{Y}\)

a linear subspace of \(\displaystyle{X}\) such that \(\displaystyle{\rm{int}(Y)\neq \varnothing}\).

Then, \(\displaystyle{X=Y}\).

Thank you r9m. Here is another idea. Suppose that \(\displaystyle{\mathcal{P}}\) (the set of polynomials) is open in \(\displaystyle{\left(C([-1,1]),||\cdot||_{\infty}\right)}\). Let \(\displaystyle{f(x)=x\,,x\in\left[-1,1\right]}\) and then \(\displaystyle{f\in\mathcal{P}\subseteq C([-1,1])}\). The...

Let \(\displaystyle{\langle{\,\,,\,\,\rangle}}\) denote the usual inner product of \(\displaystyle{\mathbb{R}^2}\). If \(\displaystyle{u\in\mathbb{R}^2}\), then \(\displaystyle{|f(u)-f(0)|=|u-0|\iff |f(u)|=|u|}\), so, \(\displaystyle{|f(u)|=|u|\,\,,\forall\,u\in\mathbb{R}^2\,,(I)}\). Now, if \(\disp...

Is the set of all polynomials open in \(\displaystyle{\left(C([-1,1])\,,||\cdot||_{\infty}\right)}\) ?

For every \(\displaystyle{n\in\mathbb{N}}\) holds \(\displaystyle{\left|n\,a_n\,\sin\,\dfrac{1}{n}\right|=n\,a_n\,\left|\sin\,\dfrac{1}{n}\right|\leq n\,a_n\,\dfrac{1}{n}=a_n}\) and \(\displaystyle{\sum_{n=1}^{\infty}a_n<\infty}\). So, the series \(\displaystyle{\sum_{n=1}^{\infty}n\,a_n\,\sin\,\dfr...

Prove that the series \(\displaystyle{\sum_{k=2}^{\infty}\dfrac{\sin\,(k\,x)}{\ln\,k}}\) and the series \(\displaystyle{\sum_{k=2}^{\infty}\dfrac{\sin\,(k\,x)}{k\,\ln\,k}}\) converge for each \(\displaystyle{x\in\left[0,2\,\pi\right]}\). Examine if the functions \(\displaystyle{x\mapsto \sum_{k=2}^{...

Using the lemma http://www.mathimatikoi.org/forum/viewtopic.php?f=28&t=1133" onclick="window.open(this.href);return false; prove that if \(\displaystyle{\left(H,\langle{,\rangle}\right)}\) is a Hilbert space and \(\displaystyle{U:H\to H}\) is a \(\displaystyle{\mathbb{C}}\) -linear and...