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?



Universal property of induced representation
You are writing a right adjoint to restriction so you have a natural $H$-module map
$$
iE\rightarrow E, \ (f(x):G\rightarrow E)) \mapsto f(1) .
$$
To cook up a map in the opposite direction, you need to use the fact that the category of $H$-modules is semisimple and choose a splitting map.


Now you use the fact the category of $G$-modules is semisimple. Because of this $E\rightarrow iE$ gives your left adjoint "locally", for this particular $E$ only. This is so called SSC (solution set condition) in Freyd's Theorem.


At this point you will need to work slightly harder. Essentially you will need to use [Freyd's Theorem](https://ncatlab.org/nlab/show/adjoint+functor+theorem). You can choose $E\rightarrow iE$ for each simple module but your task is to extend it functorially to all modules. Each module is canonically a direct sum of simples
$$
V = \oplus\_{S} Hom(S,E) \otimes S 
$$
but it does not help because it is a coporoduct and you will need products. So it boils down to understanding limits in the category of continuous modules and whether restriction preserves them. My guess is that the left adjoint (that you are looking for) exists if and only if $H$ is of finite index in $G$ (this means $H$ is open, not closed).


Here is [a recent paper that I can find](https://mathscinet.ams.org/mathscinet-getitem?mr=3977723) where a similar question has been treated. It has no answer to your question but has all the necessary techniques to attack it.


