Do you count 0 in the dimension of a vector space?
Eg. If $V_\lambda$ is the eigenspace of a certain function $f$, which has eigenvectors corresponding to $\lambda$ of $v_1, v_2, v_3$ then the basis of the eigenspace will be
$\{0,v_1, v_2, v_3\}$
Does this have dimension 3 or 4?
The dimension of a finite-dimensional vector space is defined to be the size of any basis of the space. The definition of a basis is a linearly independent spanning set.
If you add $0$ to any list of vectors the list becomes linearly dependent and hence is no longer a basis.
In the case of eigenspaces, which are spanned by eigenvectors, $0$ is not actually an eigenvector and does therefore not belong into the basis. Therefore, if $v_1, v_2, v_3$ are eigenvectors, the dimension of the corresponding eigen space is $3$.