  The Lie algebra $gl(V)$ is the Lie algebra of all endomorphisms of a
countable-dimensional complex vector space $V$. We define a tensor category of
topological representations of the Lie algebra $gl(V)$, so that $V$, its dual
and the adjoint representation $gl(V)$ are objects of this category. This makes
it an analogue of the category of finite-dimensional modules over the
finite-dimensional Lie algebra $gl(n)$. Our main result is that this category
is antiequivalent as a symmetric monoidal category to the category of tensor
representations of the Lie algebra of finitary infinite matrices.
