  A well known result from functional analysis states that any compact operator
between Hilbert spaces admits a singular value decomposition (SVD). This
decomposition is a powerful tool that is the workhorse of many methods both in
mathematics and applied fields. A prominent application in recent years is the
approximation of high-dimensional functions in a low-rank format. This is based
on the fact that, under certain conditions, a tensor can be identified with a
compact operator and SVD applies to the latter. One key assumption for this
application is that the tensor product norm is not weaker than the injective
norm. This assumption is not fulfilled in Sobolev spaces, which are widely used
in the theory and numerics of partial differential equations. Our goal is the
analysis of the SVD in Sobolev spaces. This work consists of two parts. In this
manuscript (part I), we address low-rank approximations and minimal subspaces
in H1. We analyze the H1-error of the SVD performed in the ambient L2-space. In
part II, we will address variants of the SVD in norms stronger than the
L2-norm. We will provide a few numerical examples that support our theoretical
findings.
