Infinite Dimensional Hilbert Space and Hamel Basis

Question

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.

Answer 1

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.