The linear isometries operad is an $E_\infty$ operad that acts on $U(\infty):=colim_n U(n)$.

The $n$-ary operations in that operad is the space of linear isometric maps (not necessarily surjective) from $\underbrace{\mathbb R^\infty\oplus\ldots \oplus\mathbb R^\infty}_{n}$ to $\mathbb R^\infty$.