The aim of this paper is twofold: (1) To revisit and slightly simplify the conditions on which Karl-Mikael Perfekt’s method of identifying pairs \((E_0,E)\) such that \(E_0^{\ast\ast}=E\) canonically rests upon (Section 3), and (2) To apply the framework to Lipschitz spaces over compact pointed metric spaces (Section 4).
K.-M. Perfekt’s framework [Ark. Mat. 51, No. 2, 345–361 (2013; Zbl 1283.46011)] involves a one-point compactification together with a corresponding limit process that is avoided in this paper, and so makes the setting easier to grab and probably also to handle. Let us now describe the simplified setup and statement of Perfekt’s method:
Assume that

(a)

\(X\) and \(Y\) are Banach spaces with \(X\) reflexive;

(b)

\(\mathscr{L}\subset L(X,Y)\) separates points on \(X\) and is given a topology \(\tau\) such that evaluation \(\hat{x}\) of \(T\in\mathscr{L}\) at \(x\in X\) is continuous (i.e., \(\tau\) is at least as fine as the strong operator topology).

Now let \[ E=\left\{x\in X\;:\:\|x\|:=\sup_{T\in\mathscr{L}}\|Tx\|_Y<\infty\right\} \] and observe that, by (b), \(x\to\hat{x}\) identifies \(E\) linearly and isometrically with a subspace of \(C^b(\mathscr{L},Y)\), the space of bounded continuous functions from \(\mathscr{L}\) to \(Y\). Further assume

(c)

\(E\) is closed in \(C^b(\mathscr{L},Y)\) so that both \(E\) and \(E_0=E\cap C_0(\mathscr{L},Y)\) are Banach spaces. Note that the “inverse embedding” of \((E,\|\cdot\|_\infty)\) into \((X,\|\cdot\|_X)\), by the closedness assumption, becomes continuous (due to the closed graph theorem).

(d)

\(B_{E_0}\) is norm-dense in \(B_E\) when viewed in \(X\).

{Theorem 3.1.} \(E\) is, under the conditions (a)–(d), canonically isometric to \(E_0^{\ast\ast}\).
Next, compare the above with the following result on \(M\)-structure:
{Theorem 2.1.} Let \(L\) be locally compact Hausdorff, \(Y\) be Banach, \(E\subset C^b(L,Y)\) a closed subspace with the unit ball \(B_{E_0}\) of \(E_0=E\cap C_0(L,Y)\) dense in \(B_E\) in the topology of uniform convergence on compact subsets of \(L\). Then \(E_0\) is an \(M\)-ideal in \(E\).
The next step in the paper is to apply the above two theorems in the case of Lipschitz spaces. From now on, \(M\) denotes a compact pointed metric space for which there exists a countable dense subset \(P\) such that the unit ball of \(B_{\mathrm{lip}_0(M)}\) is pointwise on \(P\) dense in \(B_{\mathrm{Lip}_0(M)}\). In this case, the second theorem above implies that \(\mathrm{lip}_0(M)\) is an \(M\)-ideal in \(\mathrm{Lip}_0(M)\) (Proposition 4.1) while the first theorem implies that \(\mathrm{Lip}_0(M)\) is the canonic bidual of \(\mathrm{lip}_0(M)\) (Theorem 4.2).
In short, for such an \(M\), \(\mathrm{lip}_0(M)\) is \(M\)-embedded with \(\mathrm{Lip}_0(M)\) its bidual. It should be noted that finding the right \(X, Y\) and \(\mathscr{L}\) for the Perfekt-setup is indeed not direct and was asked for by K.-M. Perfekt [Math. Scand. 121, No. 1, 151–160 (2017; Zbl 1434.46008)].