- #1
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Let H be a separable Hilbert space and let {e_k} be a Hilbert basis (aka total orthonormal sequence) for H. Then
[tex]|||u|||_1:=\sum_{k=1}^{+\infty}\frac{1}{2^k}|(e_k,u)|[/tex]
is a norm. If {f_k} is another Hilbert basis, we get another norm by setting
[tex]|||u|||_2:=\sum_{k=1}^{+\infty}\frac{1}{2^k}|(f_k,u)|[/tex]
How to show that these two norm are equivalent?
[tex]|||u|||_1:=\sum_{k=1}^{+\infty}\frac{1}{2^k}|(e_k,u)|[/tex]
is a norm. If {f_k} is another Hilbert basis, we get another norm by setting
[tex]|||u|||_2:=\sum_{k=1}^{+\infty}\frac{1}{2^k}|(f_k,u)|[/tex]
How to show that these two norm are equivalent?