If I start from a category theory approach I can define vector space direct sum with inclusions and direct product with projections by the universal properties in each case.
Defining the external direct sum and product shows existence in each case.
The external direct sum is a subspace of the external direct product so it makes sense to define projections for the external direct sum as the restriction to this subspace of the projections defined for the eternal direct product.
I can define the internal direct sum and show it is a direct sum with the associated inclusions.
If there are a finite number of subspaces then the internal direct sum is also a direct product with projections.
In the external or finite case I can show the relationship between the (restricted) inclusions and projections $p_\alpha \circ i_\alpha = I_\alpha$ and $\Sigma i_\alpha \circ p_\alpha (v) = v$
I'm stuck when it comes to an infinite internal direct sum (for example formed from the spans of an infinite dimensional basis). Clearly an infinite internal direct product makes no sense because of adding an infinite number of terms. So, how should one associate projections with an infinite internal direct sum ?