When is a Locally Convex Space Eberlein–Grothendieck?

The weak topology of a locally convex space (lcs) E is denoted by w. In this paper we undertake systematic study those lcs such that (E, w) (linearly) Eberlein–Grothendieck (see Definitions 1.2 and 3.1). following results obtained in our play key role: for every barrelled E, the (linearly Eberlein–Grothendieck) if only metrizable (E normable, respectively). main applications concern to continuous real-valued functions on Tychonoff X endowed with compact-open \(C_k(X)\). We prove \((C_k(X), w)\) Eberlein-Grothen—dieck) hemicompact (X compact, Besides this, show class which linearly preserves linear quotients. Various illustrating examples are provided.