Exercise 14

Bessel’s Inequality. Consider these orthonormal sets

along with the vector $\vec{v}\in\mathbb{R}^{4}$ whose components are $4$, $3$, $2$, and $1$.

• Find the coefficient $c_{1}$ for the projection of $\vec{v}$ into the span of the vector in $B_{1}$. Check that $\lvert\vec{v}\,\rvert^{2}\geq\lvert c_{1}\rvert^{2}$.

• Find the coefficients $c_{1}$ and $c_{2}$ for the projection of $\vec{v}$ into the spans of the two vectors in $B_{2}$. Check that $\lvert\vec{v}\,\rvert^{2}\geq\lvert c_{1}\rvert^{2}+\lvert c_{2}\rvert^{2}$.

• Find $c_{1}$, $c_{2}$, and $c_{3}$ associated with the vectors in $B_{3}$, and $c_{1}$, $c_{2}$, $c_{3}$, and $c_{4}$ for the vectors in $B_{4}$. Check that $\lvert\vec{v}\,\rvert^{2}\geq\lvert c_{1}\rvert^{2}+\dots+\lvert c_{3}\rvert^{2}$ and that $\lvert\vec{v}\,\rvert^{2}\geq\lvert c_{1}\rvert^{2}+\dots+\lvert c_{4}\rvert^{2}$.

Show that this holds in general: where $\{\hskip 1.00006pt\vec{\kappa}_{1},\ldots,\vec{\kappa}_{k}\hskip 1.00006pt\}$ is an orthonormal set and $c_{i}$ is coefficient of the projection of a vector $\vec{v}$ from the space then $\lvert\vec{v}\,\rvert^{2}\geq\lvert c_{1}\rvert^{2}+\dots+\lvert c_{k}\rvert^{2}$. Hint. One way is to look at the inequality $0\leq\lvert\vec{v}-(c_{1}\vec{\kappa}_{1}+\dots+c_{k}\vec{\kappa}_{k})\rvert^{2}$ and expand the $c$’s.