i found in claude Zuily's book chapter heat equation (french book) that for$\Omega$ bounded regular ,open $\subset R^{n}$ if we consider the homogenous heat equation given by:$$\partial_{t}u-\Delta u=0 \ (x,t)\in \Omega \times (0,\infty)$$ with $$u=0 \in \partial \Omega\times (0,\infty) and \ u_{0}\in L^{2}(\Omega)$$ then the solution is $$u(x,t)=\sum_{n\geq 1} a_{n}(t)e_{n}(x)$$ where $$a_{n}(t)=a_{0}\exp(-\lambda_{n}t) $$ and $\lambda_{n}$ are the eigenvalues of laplacian ; a result which also found in Brezis book my question is that in Zuily book he mentions that if
$$ u \in H^{k}(\Omega)\cap H_{0}^{1}(\Omega)$$ then $\|u\|_H^{k}$ is equivalent to
$$ \sum_{n\geq 1}\lambda^k_{n}\lvert a_{n}\rvert ^2$$
what i do is the following: case k=1 ,its clear for case k=2 ,by elliptic regularity we have that
$$\lvert u\rvert _{H^{2}}^2\leq \lvert\Delta u\rvert_{L^{2}}^2=\sum_{n\geq 1}\lambda^{2}_{n}a_{n}^{2}$$ now i assume that the result is trur for $k=l-1$ and i want to show that its true for $k=l$:and in this step i cant see how to complete my proof (and im not convinced that its the best way and may be there is better method to prove it ?)
my second question that in the same book it said that the solution $u\in C([0,\infty),L^{2}(\Omega))$ and the proof (in the book) that if $m=m_{n}$ is sequence converging to $t_{0}$
$$\|u(m)-u(t_{0})\|_{L^{2}}\to 0 \ as \ n\to \infty$$
so can i define $C([0,\infty),L^{2}(\Omega))$as the space of functions such that
$$\forall t,s\geq 0 :\|u(t)-u(s)\|_{L^{2}}\to 0\ as \ t\to s$$ in paricular i try to understand regularity of function in such space .thanks