The exact constant for the ℓ_1-ℓ_2 norm inequality

A fundamental inequality for Hilbert spaces is the ℓ_1-ℓ_2-norm inequality which gives that for any x ∈ H^n, x_1≤√(n)x_2. But this is a strict inequality for all but vectors with constant modulus for their coefficients. We will give a trivial method to compute, for each x, the constant c for which x_1=c√(n)x_2. Since this inequality is one of the most used results in Hilbert space theory, we believe this will have unlimited applications in the field. We will also show some variations of this result.