grandes-ecoles 2014 Q4b

grandes-ecoles · France · x-ens-maths__psi Matrices Matrix Norm, Convergence, and Inequality
Show that $H = l^2(\mathbb{N})$ equipped with the norm $\|.\|_2 = \sqrt{\sum_{n=0}^{+\infty} |u_n|^2}$ is a pre-Hilbert space for the inner product: $$\langle u, v \rangle = \sum_{n=0}^{+\infty} u_n v_n$$ (justify that this is indeed an inner product) then determine a Hilbert basis of $H$. 
Show that $H = l^2(\mathbb{N})$ equipped with the norm $\|.\|_2 = \sqrt{\sum_{n=0}^{+\infty} |u_n|^2}$ is a pre-Hilbert space for the inner product:
$$\langle u, v \rangle = \sum_{n=0}^{+\infty} u_n v_n$$
(justify that this is indeed an inner product) then determine a Hilbert basis of $H$.
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