  We begin a study of the representation theory of quantum continuous
$\mathfrak{gl}_\infty$, which we denote by $\mathcal E$. This algebra depends
on two parameters and is a deformed version of the enveloping algebra of the
Lie algebra of difference operators acting on the space of Laurent polynomials
in one variable. Fundamental representations of $\mathcal E$ are labeled by a
continuous parameter $u\in {\mathbb C}$. The representation theory of $\mathcal
E$ has many properties familiar from the representation theory of
$\mathfrak{gl}_\infty$: vector representations, Fock modules, semi-infinite
constructions of modules. Using tensor products of vector representations, we
construct surjective homomorphisms from $\mathcal E$ to spherical double affine
Hecke algebras $S\ddot H_N$ for all $N$. A key step in this construction is an
identification of a natural bases of the tensor products of vector
representations with Macdonald polynomials. We also show that one of the Fock
representations is isomorphic to the module constructed earlier by means of the
$K$-theory of Hilbert schemes.
