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:
\(\ds \map {\Pi_X} {\tuple {x_1, y_1} + \lambda \tuple {x_2, y_2} }\) | \(=\) | \(\ds \map {\Pi_X} {x_1 + \lambda x_2, y_1 + \lambda y_2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x_1 + \lambda x_2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {\Pi_X} {x_1, y_1} + \lambda \map {\Pi_X} {x_2, y_2}\) |
and:
\(\ds \map {\Pi_Y} {\tuple {x_1, y_1} + \lambda \tuple {x_2, y_2} }\) | \(=\) | \(\ds \map {\Pi_Y} {x_1 + \lambda x_2, y_1 + y_2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds y_1 + \lambda y_2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {\Pi_Y} {x_1, y_1} + \lambda \map {\Pi_Y} {x_2, y_2}\) |
So $\Pi_X$ and $\Pi_Y$ are linear.
We now show that they are bounded.
Let $\tuple {x, y} \in X \times Y$.
Then:
\(\ds \norm {\map {\Pi_X} {x, y} }_X\) | \(=\) | \(\ds \norm x_X\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \max \set {\norm x_X, \norm y_Y}\) | Definition of Max Operation | |||||||||||
\(\ds \) | \(=\) | \(\ds \norm {\tuple {x, y} }_{X \times Y}\) | Definition of Direct Product Norm |
and:
\(\ds \norm {\map {\Pi_Y} {x, y} }_Y\) | \(=\) | \(\ds \norm y_Y\) | ||||||||||||
\(\ds \) | \(\le\) | \(\ds \max \set {\norm x_X, \norm y_Y}\) | Definition of Max Operation | |||||||||||
\(\ds \) | \(=\) | \(\ds \norm {\tuple {x, y} }_{X \times Y}\) | Definition of Direct Product Norm |
So $\Pi_X$ and $\Pi_Y$ are bounded.
$\blacksquare$