Example 7 Let $x \geqslant 0, y \geqslant 0, z \geqslant 0, a, b, c, l, m, n$ be given positive numbers, and $a x+b y+c z=\delta$ be a constant, find
$$w=\frac{l}{x}+\frac{m}{y}+\frac{n}{z}$$

the minimum value.
By the Cauchy-Schwarz inequality, we have

So
$$\begin{array}{l} 
w \cdot \delta= {\left[\left(\sqrt{\frac{l}{x}}\right)^{2}+\left(\sqrt{\frac{m}{y}}\right)^{2}+\left(\sqrt{\frac{n}{z}}\right)^{2}\right] } \\
\cdot\left[(\sqrt{a x})^{2}+(\sqrt{b y})^{2}+(\sqrt{c z})^{2}\right] \\
\geqslant(\sqrt{a l}+\sqrt{b m}+\sqrt{c n})^{2} \\
w \geqslant \frac{(\sqrt{a l}+\sqrt{b m}+\sqrt{c n})^{2}}{\delta}
\end{array}$$

Using the condition for equality in the Cauchy-Schwarz inequality, we get \(x=k \sqrt{\frac{l}{a}}, y=k \sqrt{\frac{m}{b}}, z=k \sqrt{\frac{n}{c}}\), where \(k=\frac{\delta}{\sqrt{a l}+\sqrt{b m}+\sqrt{c n}}\), which makes \(a x+b y+c z=\delta\), and \(w=\frac{(\sqrt{a l}+\sqrt{b m}+\sqrt{c n})^{2}}{\delta}\), so
$$w_{\min }=\frac{(\sqrt{a l}+\sqrt{b m}+\sqrt{c n})^{2}}{\delta}$$