Infinite Dimensional Hilbert Space and Hamel Basis Currently I'm reading "A Course in Functional Analysis" by John B. Conway.In the section "Orthonormal Sets of Vectors and Bases" he writes "For an infinite-dimensional vector space , a basis is never a Hamel basis" without any further explanation. Can anyone throw some light on this matter?
Thanks in advance.
If $H$ is your Hilbert space and if $(e_j)_{j\in J}$ is an orthonormal basis of $H$, let $j(1),j(2),\ldots$ be a countable subset of $J$. Then$$\sum_{n=1}^\infty\frac1ne_{j(n)}\in H,$$but it cannot be expressed as a finite sum of elements of $(e_j)_{j\in J}$. Therefore, $(e_j)_{j\in J}$ is not a Hamel basis.
If H be finite dimensional space then each orthonormal bases will be Hamel bases, by Kronecker delta its easy to see. But for infinite dimensional spaces there is orthonormal bases that is not Hamel in general.