grandes-ecoles 2014 Q4d

grandes-ecoles · France · x-ens-maths__psi Matrices Matrix Norm, Convergence, and Inequality

In $H = l^2(\mathbb{N})$ equipped with the inner product $\langle u, v \rangle = \sum_{n=0}^{+\infty} u_n v_n$, let $B = (b_i)_{i \in \mathbb{N}}$ be a Hilbert basis of $H$ and $T \in \mathcal{L}(H)$. Show that the quantity (possibly infinite) $$\sum_{i=0}^{+\infty} \|T(b_i)\|^2$$ does not depend on the basis $B$. We denote $$\|T\|_2 = \sum_{i=0}^{+\infty} \|T(b_i)\|^2$$ and we set $$\mathcal{L}^2(H) = \left\{T \in \mathcal{L}(H), \quad \|T\|_2 < +\infty\right\}$$

In $H = l^2(\mathbb{N})$ equipped with the inner product $\langle u, v \rangle = \sum_{n=0}^{+\infty} u_n v_n$, let $B = (b_i)_{i \in \mathbb{N}}$ be a Hilbert basis of $H$ and $T \in \mathcal{L}(H)$. Show that the quantity (possibly infinite)
$$\sum_{i=0}^{+\infty} \|T(b_i)\|^2$$
does not depend on the basis $B$. We denote
$$\|T\|_2 = \sum_{i=0}^{+\infty} \|T(b_i)\|^2$$
and we set
$$\mathcal{L}^2(H) = \left\{T \in \mathcal{L}(H), \quad \|T\|_2 < +\infty\right\}$$

Paper Questions

Q1a Q1b Q1c Q1d Q1e Q2a Q2b Q2c Q2d Q2e Q3a Q3b Q3c Q3d Q3e Q4a Q4b Q4c Q4d Q4e Q4f Q4g Q4h Q4i