0
$\begingroup$

This is the problem 16 of Chapter 5 of Optimization be the vector space method, it is later used to prove the existence of Lagrange multipliers in equality constraint problems.

Let $g_1,g_2,\ldots, g_n$ be linearly independent linear functionals on a vector space $X$. Let $f$ be another linear functional such that for every $x$ satisfying $g_i(x)=0, i = 1,2,\ldots,n$ ,we have $f(x)=0$, show that there are constants $\lambda_1, \lambda_2, \ldots, \lambda_n$ such that $f = \sum_{i=1}^n\lambda_ig_i$.

I know it's supposed to use Hahn-Banach theorem, but I failed to figure out how.

$\endgroup$
0
$\begingroup$

I would construct vectors $x_1, \ldots, x_n$ with $g_i(x_j) = \delta_{ij}$. Then, for $x \in X$ you have $g_i(x + \alpha_1 x_1 + \ldots \alpha_n x_n) = 0$ for all $i$ and suitable $\alpha_i$. This can be used to determine $f(x)$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.