The Representer theorem on the reproducing kernel Hilbert space has the form \begin{align*} \langle K(\cdot, x_i), f\rangle_{\mathcal{H}} = f(x_i) \end{align*} where $K$ is a reproducing kernel with eigen-expansion \begin{align*} K(x, y)= \sum_{i=1}^{\infty} \lambda_i \phi_i(x) \cdot \phi_i(y) \end{align*} $f(x) = \sum_{i=1}^{\infty} c_i \phi_i(x)$ is an element in the reproducing kernel Hilbert space. $c_i$ is some real coefficient.
The last line in the above attached image shows the Representer property. I am not sure about how the 2nd equation is justified. It will be great if someone can show how the dot product is expanded, and why there is $\lambda_i$ on the denominator. Thanks in advance!