Projections on Direct Product of Normed Vector Spaces define Bounded Linear Transformations

Theorem
Let $\Bbb F$ be a subfield of $\C$.

Let $\struct {X, \norm {\, \cdot \,}_X}$ and $\struct {Y, \norm {\, \cdot \,}_Y}$ be normed vector spaces over $\Bbb F$.

Let $V = X \times Y$ be the direct product of the vector spaces $X$ and $Y$ together with induced component-wise operations.

Let $\norm {\, \cdot \,}_{X \times Y}$ be the direct product norm.

Define the maps $\Pi_X : X \times Y \to X$ and $\Pi_Y : X \times Y \to Y$ by:


 * $\map {\Pi_X} {x, y} = x$

and:


 * $\map {\Pi_Y} {x, y} = y$

for all $x \in X$, $y \in Y$.

Then $\Pi_X$ and $\Pi_Y$ are bounded linear transformations.

Proof
Let $\tuple {x_1, y_1}, \tuple {x_2, y_2} \in X \times Y$ and $\lambda \in \Bbb F$.

Then we have:

and:

So $\Pi_X$ and $\Pi_Y$ are linear.

We now show that they are bounded.

Let $\tuple {x, y} \in X \times Y$.

Then:

and:

So $\Pi_X$ and $\Pi_Y$ are bounded.