Let H be a separable infinite dimensional Hilbert space and (x_n) be a dense sequence in H. Then K(H) which is the closed linear span of rank-one projections, is the closure of the linear span of x_n[`(Ä)] x_n with rational coefficients,hence it is separable. If (e_n) is an orthonormal basis for H and for each subset S of the natural numbers N,

PS(en) = ì í î en n Î S 0 otherwise ,

then ||P_S-P_S¢|| = 1, for S ¹ S¢. Thus {P_S}_{S Î 2^N} cannot be in the closure of any countable sequence of B(H). Thus B(H) isn't separable.
Note that for x and y in H the rank one operator x[`(Ä)] y is defined by

(x Ä   y)(z) = < z,y > x.