Here is a quote from "Applied Analysis" by Hunter and Nachtergaele (p 138, Ch. 6).
Given a subset $U$ of (a Hilbert space) $H$, we define the closed linear span $[U]$ of $U$ by $$[U] = \left\{ \sum_{u \in U} c_u u \,|\, c_u \in \mathbb{C}\,\, \text{and}\,\, \sum_{u \in U} c_u u \,\,\,\text{converges unconditionally} \right\}.$$ Equivalantely, $[U]$ is the smallest closed linear subspace that contains $U$.
As far as I know, however, closedness requires mere convergence not unconditional convergence. Unconditional convergence is stronger than convergence of sequences. So, is the definition of the closed linear span of $U$ really equivalent to "the smallest closed linear subspace that contains $U$"?
I have another related question. The book proves that $E=\{ e^{inx} | n \in \mathbb{Z}\}$ is a basis of $L^2(\mathbb{T})$ where $\mathbb{T}$ is the circle of length $2\pi$. But what the book actually proves is that any function in $C(\mathbb{T})$, which is the space of continuous functions on $\mathbb{T}$, can be uniformly approximated by a sequence in $E$, and $C(\mathbb{T})$ is dense in $L^2(\mathbb{T})$ (Ch. 7). Then the book claims that $E$ is an orthonormal basis. However, the book defines an orthonromal basis of a Hilbert space as an orthonormal set whose closed linear span, which is defined above, is the whole space. Again, the definition of the basis requires unconditional convergence, but density requires just convergence.
So is it my confusion on the notion of convergence or is it a fault of the book?
The book defines convergence in the usual way of $\epsilon$-$\delta$ definition of convergence in metric spaces. The unconditional convergence is defined in the following way. An unordered sum of the indexed set $\{x_\alpha | \alpha \in I\}$ converges to $x$ if for every $\epsilon > 0$ there is a finite subset $J^\epsilon$ of $I$ such that $||\sum_{\alpha \in J}x_\alpha - x|| < \epsilon$ for all finite subset $J$ of $I$ that contain $J^\epsilon$. The convergence of an unordered sum is called unconditional convergence. Here $I$ can be countable or uncountable.