Old title: Components of unbounded endomorphism wrt Schauder basis? Changed for discoverarbility.
In finite-dimensional linear algebra, given an endomorphism $T$ and a basis $\boldsymbol{e}_i$, we can easily find a matrix representation of $T$:
$$
T(v)
= T(\sum_i v^i \, \boldsymbol{e}_i)
\overset{*}{=} \sum_i T(v^i \, \boldsymbol{e}_i)
= \sum_i v^i T(\boldsymbol{e}_i)
=: \sum_{i,j} v^i \, T_{ij} \, \boldsymbol{e}_j
$$
Let us now assume that $X$ is a separable Banach space with a Schauder basis $\{\boldsymbol{e_i}\}_{i \in \mathbb{N}}$ and $T: X \to X$ is an unbounded linear operator. If we try to replicate the previous equation, the sum will become infinite, and since for unbounded operators $T(\lim_n v_n) = \lim_n T(v_n)$ doesn't hold, the equality $(\overset{*}{=})$ above doesn't hold.
This would seem to indicate that unbounded operators don't have a coordinate representation in a Schauder basis, at least not in the same sense as bounded operators do. Or is there a simple generalization that I just don't see?
If we now consider $X$ to be a separable Hilbert space and $\boldsymbol{e}_i$ an orthonormal basis, we know that it's isomorphic to $\ell_2$ with basis:
$$
\begin{pmatrix}1\\0\\0\\\vdots\end{pmatrix},
\begin{pmatrix}0\\1\\0\\\vdots\end{pmatrix},
\begin{pmatrix}0\\0\\1\\\vdots\end{pmatrix},
\;\dots
$$
This means that either the endomorphisms on the space of „infinite column vectors“ aren't always „infinite matrices“, or that my previous conclusion about the „non-representability“ of unbounded operators was wrong. Which of these is true?
In summary, these are my questions:
- Is it possible to represent an unbounded operator using its coordinates wrt. a Schauder basis of a separable Banach/Hilbert space?
- If yes, how? If no, are there other practical ways to represent it? (Eg. using Hamel basis? Or as an integral kernel on $L^p$?)
- Are all operators on $\ell_2$ „infinite matrices“?
Best Answer
The whole field of Analysis is highly dependent on the fact that we may approximate complicated things (think a vector $x$ in your space $X$) by simpler things (think truncated sums of the representation of $x$ in your Schauder basis). These approximations are only useful because the observations we want to make (e.g. $T(x)$) respect the approximations we made (i.e., are continuous functions).
Analysis is therefore (almost) completely useless if the quantities we want to measure do not respect approximations.
Therefore, in order to study discontinuous linear transformations you might as well ignore the norm structure of $X$ and hence your Schauder basis will be devoid of any interesting properties, beyond being any old linearly independent set. In this case, as you noted, you will need a Hamel basis to describe what you see!
Disclaimer. Many discontinuous linear transformtions may be successfully studied with analytic methods because they sometimes possess hidden continuity properties, such as closed operators.