"Basic" question about Banach spaces, projections

Question

Suppose $(X,\|\cdot\|_X)$ and $(Y,\|\cdot\|_Y)$ are Banach spaces such that $Y\subset X$, $\|\cdot\|_X\leq \|\cdot\|_Y$.

Suppose further that $X=X_1\oplus X_2$. Then any $y\in Y$ can be written as $y=x_1+x_2$ for $x_1\in X_1$, $x_2\in X_2$.

Is it necessarily the case that $x_1,x_2$ are in $Y$? Phrased alternatively, is it true that $Y=(Y\cap X_1)\oplus (Y\cap X_2)$? I suspect not, but can't seem to find a counterexample. For my purposes, it would be very convenient if it was in fact true.

Note: This seems quite 'bookworky', but rest assured it is not homework!

Answer 1

Take $X=Vect(e_1,e_2)$ and $Y=Vect(e_1+e_2)$.

Answer 2

If you define $M \oplus N = \operatorname{span}(M \cup N)$, then

$$Y = Y \cap X = Y \cap (X_1 \oplus X_2) = Y \cap \operatorname{span}(X_1 \cup X_2)$$

On the other hand,

$$(Y \cap X_1) \oplus (Y \cap X_2) = \operatorname{span}\Big((Y \cap X_1) \cup (Y \cap X_2)\Big) = \operatorname{span} \Big(Y \cap (X_1 \cup X_2)\Big)$$

As noted, $Y$ can intersect $X_1 \cup X_2$ trivially, and yet intersect $\operatorname{span}(X_1 \cup X_2)$ nontrivially.