Is this a projection of a tensor state? Take the  state $|\Psi\rangle $ living in a product space of space 1 and space 2 with orthonormal bases $\varphi,\phi $ $$|\Psi\rangle=\sum_{i,j}a_ib_j|\varphi_i\rangle\otimes|\phi_j\rangle $$
Is the following action defined ?
$$
\begin{aligned}
\langle \varphi_k|\Psi\rangle &=\sum_{i,j}a_ib_j \langle \varphi_k|\varphi_i\rangle\otimes|\phi_j\rangle \\
&= \sum_{i,j}a_ib_j \delta_{ki}|\phi_j\rangle \\
\langle \varphi_k|\Psi\rangle&=\sum_{j}a_k b_j |\phi_j\rangle \\
\end{aligned}
$$
What is this called and how should you notate the result $\langle \varphi_k|\Psi\rangle$. My problem is in this notation it looks like a scalar but it should be a vector living in the space spanned by $\phi$. Does this procedure have a name and how is its proper notation ? Or is the whole thing nonsensical ?
 A: What you are proposing is connected to how one describes a projection operation on one half of a bipartite state. The quantity $\langle \varphi_k\vert\Psi\rangle$ is not a scalar - it is a vector that lives in the second subsystem and your expression for it is correct.
Assuming $\{\vert\varphi_i\rangle\}$ form an orthonormal basis, the set of projection operators $\{P_i = \vert\varphi_i\rangle\langle\varphi_i\vert\otimes I\}$ is a valid measurement since $\sum_i P_i = I\otimes I$. The probability of obtaining the result $k$ is 
$$p_k = \langle \Psi \vert P_k \vert \Psi \rangle$$ 
The post measurement state is written in terms of the quantity $\langle \varphi_k\vert\Psi\rangle$ but with an additional ket in the first subsystem. Assuming you obtained outcome $k$, the unnormalized post measurement state is 
$$P_k\vert\Psi\rangle = \vert\varphi_k \rangle \langle\varphi_{k}\vert\Psi\rangle = \sum_j a_kb_j\vert\varphi_k\rangle\otimes\vert\phi_j\rangle$$ 
Remember to normalize this state with $\frac{1}{\sqrt{p_k}}$ to obtain a valid quantum state. 
