using double quotes in bash export statement

The quotes won't hurt anything, but neither are they necessary. Assignments are processed specially by the shell. From the man page:

A variable may be assigned to by a statement of the form

          name=[value]

If value is not given, the variable is assigned the null string. All values undergo tilde expansion, parameter and variable expansion, command substitution, arithmetic expansion, and quote removal (see EXPANSION below).

Notice that word-splitting and pathname generation are not on the list in bold. These are the two types of expansion you are trying to prevent by quoting a parameter expansion, but in this context they are not performed. The same rules apply to the assignments that are passed to the export built-in command.

You must include the variable PATH inside double quotes. So that it would handle the filepaths which has spaces but without double quotes, it won't handle the filenames which has spaces in it.