This is part of an exercise in Rudin's Functional Analysis, in the chapter on Unbounded Operators. Let $H$ be a Hilbert space with orthonormal basis $\{e_n\}$. Let $X$ be the set of all finite sums $\displaystyle\sum\limits_{i=0}^n \alpha_ie_i$ where $\displaystyle\sum\limits_{i=0}^n\alpha_i=0$. The exercise is to prove that $X$ is dense in $H$. Since $\{e_i\}$ is an onb, it is clear that every $x\in H$ is a limit of finite sums of the form $\displaystyle\sum\limits_{i=0}^n \alpha_ie_i$, but how do we ensure that $\displaystyle\sum\limits_{i=0}^n \alpha_i=0$?
Asked
Active
Viewed 110 times
1
-
Hint: try to approximate the base vectors $e_k.$ – Yurii Savchuk Dec 03 '13 at 08:27
-
2Alternatively: try to compute $X^\perp.$ – Yurii Savchuk Dec 03 '13 at 08:31
-
Thank you, the second hint worked perfectly! – Arundhathi Dec 03 '13 at 08:45
-
@YuriiSavchuk you should post this as hint-like answer – Norbert Dec 03 '13 at 12:50
-
@Norbert: ok, thx. – Yurii Savchuk Dec 03 '13 at 13:27
