Say I have a Banach-space $X$ and linear (!) functionals $f,g$, and I'm trying to solve the constrained optimization problem
$$\max~f(x)\quad \mbox{s.t.}~g(x)= 0,~\Vert x\Vert\le 1.$$
Suppose I can show that a unique solution $\tilde{x}$ exists. Does this imply the existence of a Lagrange Multiplier $\lambda$ for which $\tilde{x}$ is a solution to
$$\max~\left(f(x)+\lambda g(x)\right) \quad \mbox{s.t.}~\Vert x\Vert\le 1$$
(without the constraint $g(x)=0$)? If yes, does the same hold for multiple constraints $g_1,\ldots,g_n$?
It appears that this is somewhat different from the "classical" use of Lagrange Multipliers, due to the additional constraint $\Vert x\Vert\le 1$, which restricts $x$ to the closed unit ball. Also note that the functionals are assumed to be linear, so this might allow for some simplifications. Any help is appreciated.