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Solve the following problem using Python, implementing the functions described below, one line at a time
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Description:
 def astype(self, out_dtype):
 """Return a copy of this space with new ``out_dtype``. Parameters out_dtype : Output data type of the returned space. Can be given in any way `numpy.dtype` understands, e.g. as string (``'complex64'``) or built-in type (``complex``). ``None`` is interpreted as ``'float64'``. Returns ------- newspace : `FunctionSpace` The version of this space with given data type """

        out_dtype = np.dtype(out_dtype)
        if out_dtype == self.out_dtype:
            return self

        # Try to use caching for real and complex versions (exact dtype
        # mappings). This may fail for certain dtype, in which case we
        # just go to `_astype` directly.
        real_dtype = getattr(self, 'real_out_dtype', None)
        if real_dtype is None:
            return self._astype(out_dtype)
        else:
            if out_dtype == real_dtype:
                if self.__real_space is None:
                    self.__real_space = self._astype(out_dtype)
                return self.__real_space
            elif out_dtype == self.complex_out_dtype:
                if self.__complex_space is None:
                    self.__complex_space = self._astype(out_dtype)
                return self.__complex_space
            else:
                return self._astype(out_dtype)