I don't know if this is a typo but you have two distinct methods: one that takes no parameter, and another that takes an object, so the compiler requirement is legitimate.
If this is a typo and that you have only one method, say "myFunction()", then I fear VB.Net does not act like C# by simply hiding the base interface and allowing to only implement the derived one.
But you could easily fix this by forwarding:
Namespace ns1
Public Interface IParent
Function myFunction() As Double
End Interface
End Namespace
Namespace ns2
Public Interface IDerived
Inherits ns1.IParent
Function myFunction() As Double
End Interface
End Namespace
Namespace ns3
Public Class Class1
Implements ns2.IDerived
Public Function myFunction() As Double Implements ns2.IDerived.myFunction
Return 42
End Function
Private Function myFunction1() As Double Implements ns1.IParent.myFunction
Return myFunction()
End Function
End Class
End Namespace
Module Module1
Sub Main()
Dim cp As ns1.IParent = New ns3.Class1
cp.myFunction()
Dim cd As ns2.IDerived = New ns3.Class1
cd.myFunction()
End Sub
End Module
EDIT: So was not a typo, here is the standard (good/best practice?) fix:
Public Class Class1
Implements ns2.IDerived
Public Function myFunction(ByRef obj As Object) As Double Implements ns2.IDerived.myFunction
End Function
Public Function myFunction() As Double Implements ns1.IParent.myFunction
Throw New NotImplementedException("'ns1.IParent.myFunction' has not been implemented because unicorns can't fly!")
End Function
End Class