  In this paper, we investigate the structure and representations of the
quantum group ${\mathbf{U}(\infty)}=\mathbf U_\upsilon(\frak{gl}_\infty)$. We
will present a realization for $\mathbf{U}(\infty)$, following
Beilinson--Lusztig--MacPherson (BLM) \cite{BLM}, and show that the natural
algebra homomorphism $\zeta_r$ from $\mathbf{U}(\infty)$ to the infinite
$q$-Schur algebra ${\boldsymbol{\mathcal S}}(\infty,r)$ is not surjective for
any $r\geq 1$. We will give a BLM type realization for the image
$\mathbf{U}(\infty,r):=\zeta_r(\mathbf{U}(\infty))$ and discuss its
presentation in terms of generators and relations. We further construct a
certain completion algebra $\hat{\boldsymbol{\mathcal K}}^\dagger(\infty)$ so
that $\zeta_r$ can be extended to an algebra epimorphism
$\tilde\zeta_r:\hat{\boldsymbol{\mathcal
K}}^\dagger(\infty)\to{\boldsymbol{\mathcal S}}(\infty,r)$. Finally we will
investigate the representation theory of ${\bf U}(\infty)$, especially the
polynomial representations of ${\bf U}(\infty)$.
