AbstractThis paper is on general methods of convergence and summability. We first present the general method of convergence described by free filters of $\mathbb{N} $
N
and study the space of convergence associated with the filter. We notice that $c(X)$
c
(
X
)
is always a space of convergence associated with a filter (the Frechet filter); that if X is finite dimensional, then $\ell _{\infty }(X)$
ℓ
∞
(
X
)
is a space of convergence associated with any free ultrafilter of $\mathbb{N} $
N
; and that if X is not complete, then $\ell _{\infty }(X)$
ℓ
∞
(
X
)
is never the space of convergence associated with any free filter of $\mathbb{N} $
N
. Afterwards, we define a new general method of convergence inspired by the Banach limit convergence, that is, described through operators of norm 1 which are an extension of the limit operator. We prove that $\ell _{\infty }(X)$
ℓ
∞
(
X
)
is always a space of convergence through a certain class of such operators; that if X is reflexive and 1-injective, then $c(X)$
c
(
X
)
is a space of convergence through a certain class of such operators; and that if X is not complete, then $c(X)$
c
(
X
)
is never the space of convergence through any class of such operators. In the meantime, we study the geometric structure of the set $\mathcal{HB}(\lim ):= \{T\in \mathcal{B} (\ell _{\infty }(X),X): T|_{c(X)}= \lim \text{ and }\|T\|=1\}$
HB
(
lim
)
:
=
{
T
∈
B
(
ℓ
∞
(
X
)
,
X
)
:
T
|
c
(
X
)
=
lim
and
∥
T
∥
=
1
}
and prove that $\mathcal{HB}(\lim )$
HB
(
lim
)
is a face of $\mathsf{B} _{\mathcal{L}_{X}^{0}}$
B
L
X
0
if X has the Bade property, where $\mathcal{L}_{X}^{0}:= \{ T\in \mathcal{B} (\ell _{\infty }(X),X): c_{0}(X) \subseteq \ker (T) \} $
L
X
0
:
=
{
T
∈
B
(
ℓ
∞
(
X
)
,
X
)
:
c
0
(
X
)
⊆
ker
(
T
)
}
. Finally, we study the multipliers associated with series for the above methods of convergence.
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