Let \({\mathbb {K}}\) be a non-trivially valued non-Archimedean complete field. Let \(\ell _{\infty }({\mathbb {N}}, {\mathbb {K}})\) [\(\ell _c({\mathbb {N}}, {\mathbb {K}});\)\(c_0({\mathbb {N}}, {\mathbb {K}})\)] be the space of all sequences in \({\mathbb {K}}\) that are bounded [relatively compact; convergent to 0] with the topology of pointwise convergence (i.e. with the topology induced from \({\mathbb {K}}^{{\mathbb {N}}}\)). Let X be an infinite ultraregular space and let \(C_p(X,{\mathbb {K}})\) be the space of all continuous functions from X to \({\mathbb {K}}\) endowed with the topology of pointwise convergence. It is easy to see that \(C_p(X,{\mathbb {K}})\) is metrizable if and only if X is countable. We show that for any X [with an infinite compact subset] the space \(C_p(X,{\mathbb {K}})\) has an infinite-dimensional [closed] metrizable subspace isomorphic to \(c_0({\mathbb {N}}, {\mathbb {K}})\). Next we prove that \(C_p(X,{\mathbb {K}})\) has a quotient isomorphic to \(c_0({\mathbb {N}}, {\mathbb {K}})\) if and only if it has a complemented subspace isomorphic to \(c_0({\mathbb {N}}, {\mathbb {K}})\). It follows that for any extremally disconnected compact space X the space \(C_p(X,{\mathbb {K}})\) has no quotient isomorphic to the space \(c_0({\mathbb {N}}, {\mathbb {K}})\); in particular, for any infinite discrete space D the space \(C_p(\beta D, {\mathbb {K}})\) has no quotient isomorphic \(c_0({\mathbb {N}}, {\mathbb {K}})\). Finally we investigate the question for which X the space \(C_p(X,{\mathbb {K}})\) has an infinite-dimensional metrizable quotient. We show that for any infinite discrete space D the space \(C_p(\beta D, {\mathbb {K}})\) has an infinite-dimensional metrizable quotient isomorphic to some subspace \(\ell _c^0({\mathbb {N}}, {\mathbb {K}})\) of \({\mathbb {K}}^{{\mathbb {N}}}\). If \({\mathbb {K}}\) is locally compact then \(\ell _c^0({\mathbb {N}}, {\mathbb {K}})=\ell _{\infty }({\mathbb {N}}, {\mathbb {K}})\). If \(|n1_{{\mathbb {K}}}|\ne 1\) for some \(n\in {\mathbb {N}}\), then \(\ell _c^0({\mathbb {N}}, {\mathbb {K}})=\ell _c ({\mathbb {N}}, {\mathbb {K}}).\) In particular, \(C_p(\beta D, {\mathbb {Q}}_q)\) has a quotient isomorphic to \(\ell _{\infty }({\mathbb {N}}, {\mathbb {Q}}_q)\) and \(C_p(\beta D, {\mathbb {C}}_q)\) has a quotient isomorphic to \(\ell _c({\mathbb {N}}, {\mathbb {C}}_q)\) for any prime number q.

令\（{\ mathbb {K}} \）为一个非平凡的非阿基米德完整字段。令\（\ ell _ {\ infty}（{\ mathbb {N}}，{\ mathbb {K}}）\） [ ） [ \（\ ell _c（{\ mathbb {N}}，{\ mathbb {K} }）; \）\（c_0（{\ mathbb {N}}，{\ mathbb {K}}）\） ]是\（{\ mathbb {K}} \\}中所有有界[相对紧凑；使用逐点收敛的拓扑（即\（{\ mathbb {K}} ^ {{\ mathbb {N}}} \）诱导的拓扑收敛到0] ）。令X为无限的超规则空间，令\（C_p（X，{\ mathbb {K}}）\）为从X到\（{\ mathbb {K}} \）的所有连续函数的空间具有逐点收敛的拓扑。很容易看到\（C_p（X，{\ mathbb {K}}）\）是可量化的，且仅当X是可数的。我们表明，对于任何X [具有无限紧凑子集]，空间\（C_p（X，{\ mathbb {K}}）\）具有与\（c_0（{{mathbb {N}}，{\ mathbb {K}}）\）。接下来，我们证明了\（C_P（X，{\ mathbb {K}}）\）具有商同构于\（C_0（{\ mathbb {N}}，{\ mathbb {K}}）\），当且仅如果它具有\（c_0（{\ mathbb {N}}，{\ mathbb {K}}）\）同构的互补子空间。因此，对于任何极端断开的紧凑空间X空间\（C_p（X，{\ mathbb {K}}）\）与空间\（c_0（{\ mathbb {N}}，{\ mathbb {K}}）\）没有同构。特别是，对于任何无限离散空间D，空间\（C_p（\ beta D，{\ mathbb {K}}）\）没有商同构\（c_0（{\ mathbb {N}}，{\ mathbb {K }}）\）。最后，我们探讨这问题X的空间\（C_P（X，{\ mathbb {K}}）\）具有无限维度量化商。我们证明，对于任何无限离散空间D，空间\（C_p（\ beta D，{\ mathbb {K}}）\）具有对某些子空间同构的无限维可商化商\（\ ELL _c ^ 0（{\ mathbb {N}}，{\ mathbb {K}}）\）的\（{\ mathbb {K}} ^ {{\ mathbb {N}}} \）。如果\（{\ mathbb {K}} \）是局部紧凑的，则\（\ ell _c ^ 0（{\ mathbb {N}}，{\ mathbb {K}}）= \ ell _ {\ infty}（{ \ mathbb {N}}，{\ mathbb {K}}）\）。如果\（| n1 _ {{\ mathbb {K}}} | \ ne 1 \）对于某些\（n \ in {\ mathbb {N}} \），则\（\ ell _c ^ 0（{\ mathbb { N}}，{\ mathbb {K}}）= \ ell _c（{\ mathbb {N}}，{\ mathbb {K}}）。\）特别是\（C_p（\ beta D，{\ mathbb {Q}} _ q）\）与\（\ ell _ {\ infty}（{\ mathbb {N}}，{\ mathbb {Q}} _ q）\）和\（C_p（\ beta D ，{\ mathbb {C}} _​​ q）\）的商同构\（\ ell _c（{\ mathbb {N}}，{\ mathbb {C}} _​​ q）\）对于任何质数q。