Question

Let the functional f on R² be defined by f(x)= 4x-3y.Regard R² as a subspace of R³ given by z=0 determine all linear extension of f(x) from R² to R³

Answer 1

The of functional $f(x)$ on $R^2$ :

$||f||_{R^2}=\sqrt{4^2+3^2}=5$

The norm of linear extension:

$||f_0||_{R^3}=||f||_{R^2}$

So, linear extension is:

$f_0=a_1x+a_2y+a_3z$

$\sqrt{a_1^2+a_2^2+a_3^2}=5$