On subspaces of spaces $$C_p(X)$$ isomorphic to spaces $$c_{0}$$ and $$\ell _{q}$$ with the topology induced from $$\mathbb {R}^{\mathbb {N}}$$

Abstract

AbstractThe linear space of all continuous real-valued functions on a Tychonoff space X with the pointwise topology (induced from the product space $$\mathbb {R}^X$$ R X ) is denoted by $$C_p(X).$$ C p ( X ) . In this paper we continue the systematic study of sequences spaces $$c_{0}$$ c 0 and $$\ell _{q}$$ ℓ q (for $$0<q\le \infty $$ 0 < q ≤ ∞ ) with the topology induced from $$\mathbb {R}^{\mathbb {N}}$$ R N (denoted by $$(c_{0})_p$$ ( c 0 ) p and $$(\ell _{q})_{p}$$ ( ℓ q ) p , respectively) and their role in the theory of $$C_p(X)$$ C p ( X ) spaces. For every infinite Tychonoff space X we construct a subspace F of $$C_p(X)$$ C p ( X ) that is isomorphic to $$(c_{0})_p$$ ( c 0 ) p ; if X contains an infinite compact subset, then the copy F of $$(c_{0})_p$$ ( c 0 ) p is closed in $$C_p(X)$$ C p ( X ) . It follows that $$C_p(X)$$ C p ( X ) contains a copy of $$(\ell _{q})_{p}$$ ( ℓ q ) p for every $$0<q\le \infty $$ 0 < q ≤ ∞ . We prove that for any infinite compact space X the space $$C_p(X)$$ C p ( X ) contains no closed copy of $$(\ell _{q})_{p}$$ ( ℓ q ) p for $$q\in (0, 1]\cup \{\infty \}$$ q ∈ ( 0 , 1 ] ∪ { ∞ } and no complemented copy for $$0<q\le \infty $$ 0 < q ≤ ∞ . Relation with results of Talagrand, Haydon, Levy and Odell will be also discussed. Examples and open problems will be provided.