Let $E=L^2(\mathbb{R})$ endowed with its usual topology. Let $\alpha>0$ and $\left(\varphi_n\right)_{n \in N}\subset E$ defined by $$ \varphi_n=\frac{n^{3 / 2}}{n^{2 \alpha}+x^2} $$
Give, with justification, the values of $\alpha$ for which the sequence $\left(\varphi_n\right)_{n \in v^*}$ to $0_E$.
Give, with justification, the values of Let $E=L^2(\mathbb{R})$ endowed with its usual topology. Let $\alpha>0$ and $\left(\varphi_n\right)_{n \in N}\subset E$ defined by $$ \varphi_n=\frac{n^{3 / 2}}{n^{2 \alpha}+x^2} $$
Give, with justification, the values of $\alpha$ for which the sequence $\left(\varphi_n\right)_{n \in v^*}$ to $0_E$.
Give, with justification, the values of $\alpha$ for which the sequence $\left(\varphi_n\right)_{n \in N}$ does not converge weakly to $0_E$.
I answered 2 To show that the sequence $(\psi_n)$ strongly converges in $L^2(\mathbb{R})$ to $0_{L^2(\mathbb{R})}$, that is, say that $\|\psi_n - 0_{L^2(\mathbb{R})}\|_{L^2(\mathbb{R})} \to 0$ when $n \to \infty$, where $0_{L^2(\mathbb{R})}$ is the neutral element of the space $L^2(\mathbb{R})$, one can proceed as follows:
$$ \begin{aligned} \|\psi_n - 0_{L^2(\mathbb{R})}\|_{L^2(\mathbb{R})}^2 &= \int_{-\infty}^{\infty} |\psi_n(x) - 0|^2 dx \\ &= \int_{-\infty}^{\infty} \frac{n^3}{(n^{2\alpha}+x^2)^2} dx \\ &= \frac{n^3}{n^{4\alpha}} \int_{-\infty}^{\infty} \frac{dx}{(1+(x/n^{\alpha})^ 2)^2} \\ &= \frac{1}{n^{3\alpha-3}} \int_{-\infty}^{\infty} \frac{du}{(1+u^2)^2}, \quad \text{with } u=x/n^{\alpha} \\ &\leq \frac{1}{n^{3\alpha-3}} \int_{-\infty}^{\infty} \frac{du}{1+u^2} \\ &= \frac{\pi}{n^{3\alpha-3}}, \end{aligned} $$ where we made the change of variable $u=x/n^\alpha$ and used the inequality $(1+u^2)^2 \geq (1+u^2)$ to obtain the last inequality .
As $\alpha > 1$, we have $3\alpha-3 > 0$, and therefore $\frac{\pi}{n^{3\alpha-3}} \to 0$ when $n \to \infty $. Therefore, $\|\psi_n - 0_{L^2(\mathbb{R})}\|_{L^2(\mathbb{R})} \to 0$ when $n \to \infty$ , which shows that the sequence $(\psi_n)$ strongly converges in $L^2(\mathbb{R})$ to $0_{L^2(\mathbb{R})}$.
but i'm stuck for the 3 can someone help me
$\alpha$ for which the sequence $\left(\varphi_n\right)_{n \in N}$ does not converge weakly to $\theta_E$.
