Let $K$ be a $n$-dimensional vector subspace of $\mathbb R^N$. Then we have the surjective canonical homeomorphism $$f: \mathbb R^N \to \mathbb R^N/K, \quad x \mapsto x+K.$$
We have $\operatorname{ker} f = K$. By rank–nullity theorem, $\dim (\mathbb R^N/K) = N-n$. There exists $H \subseteq \mathbb R^N$ such that $\mathbb R^N/K \cong H$. This implies $H$ is a $(N-n)$-dimensional vector subspace of $\mathbb R^N$. Also, $\mathbb R^N \cong (\mathbb R^N/K) \oplus_e K$.
I would like to ask if we have $\mathbb R^N = H \oplus_i K$. Here $\oplus_e$ and $\oplus_i$ are the external and internal direct sums respectively.
Since $H$ was chosen extremely generally, it's quite easy to arrange for this to fail to be an internal direct sum. In $\Bbb R^2$, for example, take $K=H=\{(x,0)\colon x\in\Bbb R\}\cong \Bbb R^1$.