AbstractLet E be a Banach space with dual space $E^{*}$
E
∗
, and let K be a nonempty, closed, and convex subset of E. We generalize the concept of generalized projection operator “$\Pi _{K}: E \rightarrow K$
Π
K
:
E
→
K
” from uniformly convex uniformly smooth Banach spaces to uniformly convex uniformly smooth countably normed spaces and study its properties. We show the relation between J-orthogonality and generalized projection operator $\Pi _{K}$
Π
K
and give examples to clarify this relation. We introduce a comparison between the metric projection operator $P_{K}$
P
K
and the generalized projection operator $\Pi _{K}$
Π
K
in uniformly convex uniformly smooth complete countably normed spaces, and we give an example explaining how to evaluate the metric projection $P_{K}$
P
K
and the generalized projection $\Pi _{K}$
Π
K
in some cases of countably normed spaces, and this example illustrates that the generalized projection operator $\Pi _{K}$
Π
K
in general is a set-valued mapping. Also we generalize the generalized projection operator “$\pi _{K}: E^{*} \rightarrow K$
π
K
:
E
∗
→
K
” from reflexive Banach spaces to uniformly convex uniformly smooth countably normed spaces and study its properties in these spaces.
Local Lipschitz continuity of the metric projection operator
