The Sobolev embedding theorem as stated in my notes says that if we have $k > l + d/2$ then we can continuously extend the inclusion $C^\infty(\mathbb T^d) \hookrightarrow C^l(\mathbb T^d)$ to $H^k(\mathbb T^d) \hookrightarrow C^l(\mathbb T^d)$.
We define $H^k$ to be the closure of $C^\infty$ with respect to the Sobolev norm, see my previous question for the definition.
What I'm confused about is, why we need the condition $k > l + d/2$. What exactly does it give us? If $H^k$ is the closure of $C^\infty$ we already get that if $T$ is any continuous linear operator $C^\infty \to C^l$ we can extend it to all of $H^k$. What am I missing? Thanks for your help.