2 I have deleted a unrellevant question.

Let $\omega_1$ be smallest uncountable ordinal. I am trying to understand the possible "large" subspaces of $C[0,\omega_1]$, namely those which are isomorphic to the whole space. Are there uncomplemented subspaces Therefore I have the followin question:

Does every subspace of $C[0,\omega_1]$ isomorphic to $C[0,\omega_1]$? If yes, must they C[0,\omega_1]$contain a complemented copies of$C[0,\omega_1]$? copy isomorphic to itself? The only (complemented) examples that I can find"construct by hand", excluding the finite-codimensional ones, are of the form$\mbox{cl lin}(\mathbf{1}_{[0,\gamma\sigma]}\colon lin}(\mathbf{1}_{[0,\gamma{\sigma}]}\colon \sigma\leq \omega_1)$where$(\gamma_\sigma)_{\sigma<\omega_1}$is increasing long sequence of limit ordinals and$\sigma_{\omega_1}=\omega_1$(note that the family$({\mathbf{1}_{[0,\alpha]}\colon \alpha\leq \omega_1})$forms the long Schauder basis for$C[0,\omega_1]$.C[0,\omega_1]$).

Thank you, T.

1

# Subspaces isomorphic to C[0,omega_1]

Let $\omega_1$ be smallest uncountable ordinal. I am trying to understand the possible "large" subspaces of $C[0,\omega_1]$, namely those which are isomorphic to the whole space. Are there uncomplemented subspaces of $C[0,\omega_1]$ isomorphic to $C[0,\omega_1]$? If yes, must they contain complemented copies of $C[0,\omega_1]$? The only (complemented) examples that I can find, excluding the finite-codimensional ones, are of the form

$\mbox{cl lin}(\mathbf{1}_{[0,\gamma\sigma]}\colon \sigma\leq \omega_1)$

where $(\gamma_\sigma)_{\sigma<\omega_1}$ is increasing long sequence of limit ordinals and $\sigma_{\omega_1}=\omega_1$ (note that the family $({\mathbf{1}_{[0,\alpha]}\colon \alpha\leq \omega_1})$ forms the long Schauder basis for $C[0,\omega_1]$.

Thank you, T.