Let $X_1, X_2, \dots, X_n, Y$ be normed linear spaces, endowed with the norms $\|x_i\|_{X_i}$ and $\|y\|_Y$, respectively. Set $^nX = \prod_{i=1}^n X_i$, endowing it with the norm $\|x\| = \sup_{i}\{\|x_i\|_{X_i}\}$. Let $\mathscr{L}(A, B)$ denote the set of all linear continuous mappings between any normed linear spaces $A$ and $B$, and let $\mathscr{L}_n(X, Y)$ denote the set of all multilinear/$n$-linear mappings from $X$ to $Y$.
It is a known fact that we have the following isometric isomorphism:
$$\mathscr{L}_n(^nX, Y) \cong \mathscr{L}(X_1, \mathscr{L}(X_2, \dots, \mathscr{L}(X_n, Y)\dots)):= Z_n. \tag{1}$$
The primary application of this fact is the identification of higher Fréchet derivatives with multilinear mappings, as such: if $u \in Z_n$, we define $g \in \mathscr{L}_n(^nX, Y)$ by setting
$$g(x_1, \dots, x_n) = u(x_1)(x_2)...(x_n). \tag{2}$$
In fact, using this identification we can prove (1).
Now, given that the case is true for $n=2$, i.e. that
$$\mathscr{L}_2(X_1 \times X_2, Y) \cong \mathscr{L}(X_1, \mathscr{L}(X_2, Y)),$$
Dieudonné claims in his "Foundations of Modern Analysis" that one can prove (1) using induction on $n$ (the proper quote with my notation is "By induction on $n$, it follows that $\mathscr{L}(^kX, Y)$ can be naturally identified [...] to $Z_n$"), which I do not doubt, but I just can't figure it out. Here's an attempt to a solution:
We know that the case is true for $n=2$, so assume that the case is true for $n=k$. Set $F = \mathscr{L}(X_{k+1}, Y)$. Then by the induction hypothesis
\begin{align} Z_{k+1} & = \mathscr{L}(X_1, \mathscr{L}(X_2, \dots, \mathscr{L}(X_k,F) \dots)), \\ & \cong \mathscr{L}_k(^kX, F), \\ & = \mathscr{L}_k(^kX, \mathscr{L}(X_{k+1}, Y)) \tag{3}. \end{align}
(The isomorphism on the second line is the reason I call it an "attempt" because I am not sure I can do this, given that the induction hypothesis should only be true for $Y$ and not any normed vector space?) We can then show separately that the set in (3) is isometrically isomorphic to $\mathscr{L}_{k+1}(^{k+1}X, Y)$, the proof which I will not expand on here.
But firstly, I am not sure this proof works, and secondly it seems clunky given that I need a separate proof of another isometric isomorphism.
Maybe I am misunderstanding Dieudonné's comment, and in fact, he means that one just needs to inductively produce the identification (2), but if not, I am very curious if there is another, "neater" way to prove it than I have suggested above (given that the proof is valid).
To summarise, I really have two questions: 1. Is my proof correct? 2. If not, is there another way to prove (1) using induction? 3. If yes, is there a neater way to prove it than I have done above?