Sectional boundedness (AB) and functional sectional convergence (FAK) in FK-spaces (=locally convex, metrizable and complete spaces of complex sequences x = ({x_k})_{k = 1}^∞ ) are considered with emphasis on sums E + F = \left{ {x:x = a + b, a \in E, b \in F} \
ight} of FK-spaces E and F.
Notations and definitions:
Let {δ ^k} = (δ _n^k)_{n = 1}^∞, δ _n^n = 1 if n \
e k. {P_n}x = ∑\
olimits_{k = 1}^n {{x_k}{δ ^k}} is the n-th section of x. Let E be an FK-space, E the space of linear and continuous functionals on E. Let {P_n}x \in E for every n = 1, 2, … . x has AB in E if \left{ {{P_n}x} \
ight} is a bounded set in E, x has AK in E if {\lim _n}{P_n}x = x in E, x has FAK in E, if {\lim _n}\left( {{P_n}x} \
ight) exists for every n = 1, 2, …, x has AD in E if x \in {\bar \varphi ^E} (=the closure of φ in E), where φ is the space of x with only finitely many {x_k} \
e 0. The spaces {E
AB}, {E
Ak}, {E
FAK}, {E
AD} consisting of all x which have respectively AB, AK, FAK, AD in E, are FK-spaces with appropriate topologies. Let {E_f} = \left{ {x:{x_k} = f({δ ^k}){ for some }f \in E'{ if }{δ ^k} \in E} \
ight}, {E^r} = \left{ {x:{{\sup }_n}\left| {∑\
olimits_{k = 1}^n {{x_k}yk} } \
ight| < ∞ { for every }y \in E} \
ight}. If A and B are sets of complex sequences, then (A → B) = \left{ {x:({x_k}{y_k}) \in B{ for every }y \in A} \
ight}.
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