Coordinate subspace ($E_{\mathcal{B}}=\mathrm{span}\{e_i:i\in \mathcal{B}\}$)

For a chosen basis $(e_1,\dots ,e_n)$, the vector subspace spanned by a specified collection of basis vectors. In the standard basis of Euclidean space, its vectors have zero coordinates outside $\mathcal{B}$. The block coordinate gradient is the orthogonal projection of the full gradient onto this subspace. The fixed coordinate subspace consisting of the first $k$ coordinates is a particular example.