Let $\mathbf V$ be a banach space and $\mathbf V = \mathbf E \oplus \mathbf F$ a decomposition into closed complemented subspaces . Let $$ G(\mathbf V; \mathbf E) = \left\{ g \in \mathsf{GL}(\mathbf V) : g(\mathbf E) = \mathbf E \right\} $$ This is a closed lie subgroup of $\mathsf{GL}(\mathbf V)$ (in the sense: both a submanifold and lie group). Is there some (smooth) mapping
$$ p : \mathsf{GL}(\mathbf V) \longmapsto G(\mathbf V; \mathbf E)$$ such that $p(g) = g$, for all $g \in G(\mathbf V; \mathbf E) $? $p$ need not be a lie group morphism.