Proving $\|u\|_{H^k}\sim\sum_{n=1}^\infty\lambda_k\lvert a_n\rvert^2$, where $u$ solves heat equation $u_t=\Delta u$ and $u=\sum_{n=1}^\infty a_ne_n$

Question

In Claude Zuily's book Distribution and partial differential equations, it's mentioned that: $$\|u\|_{H^{k}}\sim\sum_{n=1}^{\infty}\lambda_{k}\lvert a_{n}\rvert^{2} \tag1$$ where $u$ is the solution of heat equation: $u_{t}=\Delta u$ and $u=\sum_{n=1}^{\infty}a_{n}e_{n}$

I don't see how to prove that? I mean how to prove $(1)$.

Thanks