Let $H$ be a closed subgroup of the compact Lie group $G$. Let $E$ be a continuous representation of $H$. In the book *"Representations of compact Lie groups"* by Bröcker and Dieck the induced representation of $E$ is defined as the vector space $iE$ of all continuous functions $f:G\to E$ satisfying $f(g\cdot h)=h^{-1}f(g)$ for all $g\in G$ and $h\in H$. They show that as in the finite case this construction satisfies the Frobenius reciprocity theorem.

Now I wonder whether this construction also satisfies the universal property that we know from the case of finite groups (or, more generally, finite index), i.e., my question is whether the following is true:

There exists an $H$-linear map $j:E\to iE$ such that for all $H$-linear maps $g:E\to E'$ to a $G$-module $E'$ there is a unique map $G$-linear map $g':iE\to E'$ such that $g'\circ j=g$.

Moreover, is $g'$ continuous if $g$ is? If the answer is "No", is there a better notion of induced representation that makes this true? Or does it help when we restrict to unitary representations?