diff --git a/arraycontext/impl/pytato/fake_numpy.py b/arraycontext/impl/pytato/fake_numpy.py
index 21dc71ed..a9158fda 100644
--- a/arraycontext/impl/pytato/fake_numpy.py
+++ b/arraycontext/impl/pytato/fake_numpy.py
@@ -240,6 +240,6 @@ def absolute(self, a):
         return self.abs(a)
 
     def vdot(self, a: Array, b: Array):
-
         return rec_multimap_array_container(pt.vdot, a, b)
+
     # }}}
diff --git a/test/test_utils.py b/test/test_utils.py
index 3b74a42f..eeef7723 100644
--- a/test/test_utils.py
+++ b/test/test_utils.py
@@ -27,7 +27,7 @@
 THE SOFTWARE.
 """
 import logging
-from typing import cast
+from typing import Optional, Tuple, cast  # noqa: UP035
 
 import numpy as np
 import pytest
@@ -63,7 +63,7 @@ def test_dataclass_array_container() -> None:
     class ArrayContainerWithOptional:
         x: np.ndarray
         # Deliberately left as Optional to test compatibility.
-        y: np.ndarray | None
+        y: Optional[np.ndarray]  # noqa: UP045
 
     with pytest.raises(TypeError, match="Field 'y' union contains non-array"):
         # NOTE: cannot have wrapped annotations (here by `Optional`)
@@ -76,7 +76,8 @@ class ArrayContainerWithOptional:
     @dataclass
     class ArrayContainerWithTuple:
         x: Array
-        y: tuple[Array, Array]
+        # Deliberately left as Tuple to test compatibility.
+        y: Tuple[Array, Array]  # noqa: UP006
 
     with pytest.raises(TypeError, match="Typing annotation not supported on field 'y'"):
         dataclass_array_container(ArrayContainerWithTuple)