A subset A of a locally convex space E is called (relatively) sequentially complete if every Cauchy sequence \(\left\{ x_{n}\right\} _{n=1}^{\infty }\) in E contained in A converges to a point \(x\in A\) (a point \(x\in E\)). Asanov and Velichko proved that if X is countably compact, every functionally bounded set in \(C_{p}\left( X\right) \) is relatively compact, and Baturov showed that if X is a Lindelöf \(\Sigma \)-space, each countably compact (so functionally bounded) set in \( C_{p}\left( X\right) \) is a monolithic compact. We show that if X is a Lindelöf \(\Sigma \)-space, every functionally bounded (relatively) sequentially complete set in \(C_{p}\left( X\right) \) or in \(C_{w}\left( X\right) \), i. e., in \(C_{k}\left( X\right) \) equipped with the weak topology, is (relatively) Gul’ko compact. We get some consequences.

子集甲一个局部凸空间的ë被称为（相对）顺序地完成，如果每个柯西序列\（\左\ {X_ {N} \右\} _ {N = 1} ^ {\ infty} \）在ë包含在甲收敛到一个点\（X \在A \） （点\（X \于E \） ）。Asanov和Velichko证明，如果X是非常紧凑的，则\（C_ {p} \ left（X \ right）\）中的每个有界集都是相对紧凑的，而Baturov表明，如果X是Lindelöf \（\ Sigma \）空间，每个空间都在\（C_ {p} \ left（X \ right）\）中设置是整体式紧凑型。我们证明，如果X是Lindelöf \（\ Sigma \） -空间，则每个在（（C）{C_ {p} \ left（X \ right）\）或\（C_ {w} \ left（X \ right）\），i。例如，在配备了弱拓扑的\（C_ {k} \ left（X \ right）\）中，（相对）Gul'ko紧凑。我们得到一些后果。