In Nielsen & Chuang, shouldn't $P=\sum_{i=1}^k|i\rangle\!\langle i|$ in (2.35) equal the identity?

Question

Nielsen and Chuang define Projectors as:

An operator $A$ whose adjoint is $A$ is known as a Hermitian or self-adjoint operator. An important class of Hermitian operators is the projectors. Suppose $W$ is a $k$-dimensional vector subspace of the $d$-dimensional vector space $V$. Using the Gram-Schmidt procedure it is possible to construct an orthonormal basis $\vert 1\rangle,\ldots ,\vert d\rangle$ for $V$ such that $\vert 1\rangle,\ldots ,\vert k\rangle$ is an orthonormal basis for $W$. By definition, $$P=\sum_{i=1}^k \vert i\rangle\langle i\vert\tag{2.35}$$

I have a basic doubt - (I think I am missing something simple, so please excuse my limited understanding): Since it is a sum over all orthonormal basis for $W$, should this not be the Identity, $\mathbb I$, according to the Completeness Relation?

Answer 1

$P$ is acting on the space $V$, projecting onto the subspace $W$. Yes, if it only acted on the subspace $W$, it would be identity, but it is acting on a larger space.

For example, think about a qubit, where $V$ is spanned by the basis states $\{|0\rangle,|1\rangle\}$. You can define a projector $P=|0\rangle\langle 0|$ which projects onto the space $W$ which, in this instance, is a single state $|0\rangle$.