the set of compact operators on $H$ is nonunital How to prove $K(H)$ is a nonunital ,where $K(H)$ is the set of compact operators on $H$,$H$ is a infinite dimensional Hilbert space? Can anyone give me some hints?Thanks

For a projection $P\in B(H)$ to be compact, it has to be finite-rank. This is because its rank is $PH$, and the unit ball of a subspace will be compact if and only if said subspace is finite-dimensional. 
Since $I$ is also a projection, it can only be compact when its image is finite-dimensional; that is, when $H$ is finite-dimensional. 
