Atlantic Ocean has a deeply/deep indented coast line which facilitates trading and other mercantile activities.
My understanding says deeply would be the appropriate choice. But I'm not sure.
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They mean two different things; which one you want depends on which meaning you want.
As a side note, it should be written as The Atlantic Ocean rather than just Atlantic Ocean
The Atlantic Ocean has a deeply indented coast line which facilitates trading and other mercantile activities.
This indicates a greater magnitude of indentation of the coast line.
The Atlantic Ocean has a deep indented coast line which facilitates trading and other mercantile activities.
This would likely be written with a comma (deep, indeted coast line), and means that the coast line is both deep and indented.
"Deeply" is an adverb. It modifies a verb or an adjective. "He thought deeply about the problem." "He stood on the edge of a deeply plunging chasm."
"Deep" is normally an adjective. It modifies a noun. "He is a deep thinker." "He stood on the edge of a deep chasm."
"Deep" can also be used as an adverb, though this is generally limited to some specific contexts, almost an idiom. "The squad advanced deep into enemy territory." "Still waters run deep."
If you wanted to say that the coastline is indented, and that the waters on the coast are deep, you would say "The deep, indented coastline ..." In that case, you would have two adjectives modifying the same noun.
But it appears that what you want to say here is that the indentations go very far into the land. That is, you want to modify the adjective "indented" with an adverb indicating "deep". In that case you would say "The deeply indented coastline ..."
Both deep and deeply are used as adverbs, but deeply can also mean "very; very much." (Deep is also used as adjective.)
In your case, since you are not using any comma after deep/deeply, the word to use is probably deeply.
However, states with coastlines that are "deeply indented" or "fringed with islands" are allowed to establish straight baselines in order to simplify the task of defining maritime jurisdiction.
Atlantic Ocean has a deeply indented coastline which facilitates trading and other mercantile activities.