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.

$\blacksquare$