Is there a solution to $Tx=y$ such that $\|x\|$ is minimal? I recently used the method of least norms to solve an underdetermined system of linear equations for a problem at work.  This got me thinking, if I were to think about this more generally, does such a solution exits?
That is, if $T$ is an operator on an inner-product space $V$, and if $x,y\in V$, is it possible to find a solution to 
$$Tx=y$$
such that $\|x\|$ is minimal?
I know that this doesn't exactly line up with the problem at work because there I was going from $\mathbb{R}^n$ to $\mathbb{R}^m$, but I thought it may be simpler to consider the case of an operator on an inner product space than a linear map between two different spaces.
I want to write this up to present it to a class, so if you know of any good sources where I might read up on this in depth, I would appreciate that as well.

For a solution to exist we must require that $y$ is in the range of $T$.
Next we note that adding an element of the null space $\mathcal{N}$ of $T$
to $x$ does not change things. Thus the allowable $x$ are
\begin{equation*}
x=u+v,\;u\perp \mathcal{N},\;v\in \mathcal{N}
\end{equation*}
Note that $u$ is unique since if $u_{1}$ and $u_{2}$ are both $\perp
\mathcal{N}$ then their difference is in $\mathcal{N}$. Now
\begin{equation*}
\parallel x\parallel ^{2}=\parallel u\parallel ^{2}+\parallel v\parallel ^{2}
\end{equation*}
which is minimal for $v=0$.
