Let $(V, \lVert \cdot \rVert)$ be a normed vector space, and let $X, Y \subseteq V$ be linear subspaces such that $X + Y$ is a direct sum (that is, $X \cap Y = \{0\}$). Since the sum is direct, every element of $X + Y$ can be uniquely written as $x + y$ for a unique $x \in X$ an a uniuqe $y \in Y$, so we can define the corresponding projection maps $\pi_X \colon X + Y \twoheadrightarrow X$ and $\pi_Y \colon X + Y \twoheadrightarrow Y$, which are linear.
Are the projection maps continuous (with respect to the norm)?
I think I have a proof for this for the finite-dimensional case (below), but I'm not sure how to extend this to the general case.
Edit: what about when $V$ is Banach and $X$ and $Y$ are closed?
Suppose $V$ is finite-dimensional.
Note that the following defines a norm on $X + Y$:
$$ \lVert x + y \rVert_{X + Y} := \lVert x \rVert + \lVert y\rVert $$
Since in finite dimensional spaces all norms are equivalent, we can assume without loss of generality that the norm $\lVert \cdot \rVert$ restricted to $X + Y$ satisfies this property.
We will prove that $\pi_X$ is continuous at any point $p = x + y \in X + Y$.
Indeed, let $\varepsilon > 0$. Then, if we set $ \delta := \varepsilon$, for all $q = x' + y' \in B_\delta(p)$ we have:
$$ \lVert \pi_X(q) – \pi_X(p) \rVert = \lVert x' – x \rVert \le \lVert x' – x \rVert + \lVert y' – y \rVert = \lVert (x' – x) + (y' – y) \rVert = \lVert q – p \rVert < \delta = \varepsilon $$
Best Answer
In finite-dimensional spaces, all linear maps are continuous. Hence are projections on linear subspaces.
See here for counterexamples in the infinite-dimensional case.