Finding the infimum of summable sequences in a decomposition

Question

Let $\{e_k\}_{k=1}^2$ denote the canonical orthonormal basis in $\mathbb{C}^2$. Put $\{f_k\}_{k=1}^4 = \{e_1, e_2, e_1-e_2, e_1 + e_2\}$. Clearly we have a representation of $e_1$ in terms of $\{f_k\}_{k=1}^4$. I am interested in finding $$m = \inf\left\{\sum_{k=1}^4 |c_k|: e_1 = \sum_{k=1}^4 c_kf_k\right\}.$$ Clearly $m \leq 1$ since we can take $c_1 = 1$ and $c_2 = c_3 = c_4 = 0$. However, I believe that $m = 1$. To show this assume that we have $$e_1 = \sum_{k=1}^4 c_kf_k,$$ for some $\{c_k\}_{k=1}^4$. How can I show that $$\sum_{k=1}^4 |c_k| \geq 1?$$

Answer 1

Here is a way to do it!

You can first solve the linear system of equations: that will let you find the coefficients for linear combinations of your generators that result in $e_1$.

For that, first solve for $e_1$, for example $e_1=f_1$, and then solve the homogeneous equation to see all possible solutions.

I got the following solution for the homogeneous equation: $$0 = \alpha (f_1 + f_2 - f_3) + \beta(f_1 - f_2 - f_4) \forall \alpha, \beta \in \mathbb{C}.$$

Now note that if $e_1 = \sum_{c=1}^{4} c_i f_i$ then we have that we can equivalently suppose that the $c_i$ are of the form: $$c_1 = 1 + \alpha - \beta, \\ c_2 = \alpha - \beta, \\ c_3 = -\alpha, \text{ and}\\ c_4 = - \beta.$$

Now you can replace each $c_i$ by these expressions and minimize over $\alpha$ and $\beta$ instead.

However, note that $$|c_1| + |c_2| + |c_3| + |c_4| \geq \\ (1 - |\alpha| - |\beta|) + |\alpha - \beta| + |\alpha| + |\beta| = \\ 1 + |\alpha - \beta| \geq 1$$ which in combination with the fact that the sum is $1$ when $\alpha=\beta=0$ proves your affirmation is true.