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

Topological tensor representations of gl(V) for a space V of countable dimension