$ \newcommand{\ket}[1]{\left| #1 \right>} \newcommand{bra}[1]{\left< #1 \right|} $Talking about the partial measurement the professor defines the state $\ket \psi$ to be
$$\ket{\psi} = \sum_{i,j} a_{ij} \ket{e_i} \otimes \ket{f_j} $$
where $\ket{e_i} \in V$ and $\ket{f_j} \in W$ are orthonormal bases. Then he rewrites the state $\ket \psi$ as
$$\ket \psi = \sum_i \ket {e_i} \otimes \ket{w_i}$$
where $\ket{w_i}= \sum_j a_{ij}\ket{f_j}$. I'm find until now. However he does the following:
$$\bra{e_i}\sum_j \ket{e_j} \otimes \ket{w_j}$$
and says, I quote, "You should understand this equation as $\bra{e_i}$ only talks to $\ket{e_j}$. You could have written the $\bra{e_i}$ in different ways maybe you could have written $\bra{e_i} \otimes \mathbf 1$" and he says that writing $\bra{e_i} \otimes \mathbf 1$ is a little strange because $\bra{e_i}$ is a bra and $\mathbf 1$ is an operator.
Assuming that what we mean by the bra $\bra{e_i}$ is $\bra{e_i} \otimes \mathbf 1$, I have difficulty understanding what the following mathematical object is:
$$\bra{e_i} \otimes \mathbf 1 \sum_j \ket{e_j} \otimes \ket{w_j} = \sum_j \delta_{ij} \otimes \ket{w_j}$$
I can think of $\delta_{ij}$ as a number but then I don't know what $1 \otimes \ket{w_i}$ (note that $1$ is a number in this expression) should mean.
If on the other hand, I think of $\delta_{ij}$ as a tensor then I cannot simplify this any further and I should write the last expression as:
$$\delta_i^{\;j} \otimes \ket{e_j}$$
where there is a funny summation over $j$. Either way I cannot reduce it any further in order to get to the equation that he has at the very last which is the following:
$$\bra{e_i}\sum_j \ket{e_j} \otimes \ket{w_j} = \ket{w_i}$$
I don't understand how we have gone from a space with dimension $v\cdot w$, where $\mathrm{dim}(V)=v$ and $\mathrm{dim}(W)=w$, to a space with dimension $w$. What is the meaning of $s\otimes \ket{e_i}$, where $s$ is a scalar, if such an object really exists and lastly is the mathematics behind the above calculation is correct?