| Summary: | Abstract Let 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 “ Π K : E → K $\Pi _{K}: E \rightarrow K$ ” 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 Π K $\Pi _{K}$ 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 Π K $\Pi _{K}$ 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 Π K $\Pi _{K}$ in some cases of countably normed spaces, and this example illustrates that the generalized projection operator Π K $\Pi _{K}$ in general is a set-valued mapping. Also we generalize the generalized projection operator “ π K : E ∗ → K $\pi _{K}: E^{*} \rightarrow K$ ” from reflexive Banach spaces to uniformly convex uniformly smooth countably normed spaces and study its properties in these spaces.
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