Vrftjkry

I'm not sure how to prove this without just using the fact that is equivalent to say that a linear functional is continous.

For

$vec v in Bbb R^n, tag 1$

we may write

$vec v = displaystyle sum_1^n v_i vec e_i, tag 2$

where $vec e_i$, $1 le i le n$ is a basis of $Bbb R^n$, orthonormal in the Euclidean inner product $langle cdot, cdot rangle_2$ on $Bbb R^n$, which defines the Euclidean norm norm $Vert cdot Vert_2 = langle cdot, cdot rangle^{1/2}$ on $Bbb R^n$ (see note below); then with

$phi: Bbb R^n to Bbb R tag 3$

linear, we have

$phi(vec v) = phi left (displaystyle sum_1^n v_i vec e_i right ) = displaystyle sum_1^n v_i phi(vec e_i); tag 4$

thus

$vert phi(vec v) vert = left vert displaystyle sum_1^n v_i phi(vec e_i) right vert le displaystyle sum_1^n vert v_i phi(vec e_i) vert = sum_1^n vert v_i vert vert phi(vec e_i) vert; tag 5$

let

$M = max {vert phi(vec e_i) vert, ; 1 le i le n }; tag 6$

then

$vert phi(vec v) vert le displaystyle sum_1^n vert v_i vert vert phi(vec e_i) vert le M sum_1^n vert v_i vert; tag 7$

also,

$vert v_i vert le sqrt { displaystyle sum_1^n v_i^2 }, 1 le i le n; tag 8$

whence

$displaystyle sum_1^n vert v_i vert le n sqrt { displaystyle sum_1^n v_i^2 }, tag 9$

so that (7) becomes

$vert phi(vec v) vert le M n sqrt { displaystyle sum_1^n v_i^2 } = MnVert vec v Vert_2, tag {10}$

which shows that $phi$ is bounded in the Euclidean norm $Vert cdot Vert_2$ on $Bbb R^n$, with bound $Mn$.

Nota Bene: The definition of Euclidean norm $Vert cdot Vert_2$ on $Bbb R^n$ used here is the usual one defined in terms of the Euclidean inner product $langle cdot, cdot rangle_2$, where if

$vec w = displaystyle sum_1^n w_i vec e_i, tag{11}$

we have

$langle vec v, vec w rangle_2 = displaystyle sum_1^n v_i w_i, tag{12}$

from which

$Vert vec v Vert_2 = sqrt{langle vec v, vec v rangle_2} = sqrt {displaystyle sum_1^n v_i^2}; tag{13}$

the vectors $vec e_i$ are chosen orthonormal with respect to this basis:

$langle vec e_i, vec e_j rangle = delta_{ij}, ; 1 le i, j le n. tag{14}$

End of Note.