fix a bunch of typos and add some spaces after commas

Author: fchapoton

src/sage/algebras/fusion_rings/f_matrix.py

@@ -1991,7 +1991,7 @@ def _get_explicit_solution(self, eqns=None, verbose=True):
 
     def find_orthogonal_solution(self, checkpoint=False, save_results='', warm_start='', use_mp=True, verbose=True):
         r"""
-        Solve the the hexagon and pentagon relations, along with
+        Solve the hexagon and pentagon relations, along with
         orthogonality constraints, to evaluate an orthogonal F-matrix.
 
         INPUT:

src/sage/algebras/hecke_algebras/cubic_hecke_algebra.py

@@ -352,7 +352,7 @@ def matrix(self, subdivide=False, representation_type=None, original=False):
             [(-2*a + u)*b - 2*a^2 + 2*u*a - v     b    0]
             [                               b     1    a]
 
-        using the the ``representation_type`` option::
+        using the ``representation_type`` option::
 
             sage: CHA3.<c0, c1> = algebras.CubicHecke(3)     #  optional gap3
             sage: chevie = CHA3.repr_type.SplitIrredChevie   #  optional gap3
@@ -364,7 +364,7 @@ def matrix(self, subdivide=False, representation_type=None, original=False):
             [            b             0]
             [a^2 - u*a + v    -b - a + u]
 
-        using the the ``original`` option::
+        using the ``original`` option::
 
             sage: c0mo = c0.matrix(original=True)
             sage: c0mo_ch = c0.matrix(representation_type=chevie, original=True) #  optional gap3

src/sage/homology/homology_vector_space_with_basis.py

@@ -1253,7 +1253,7 @@ def _acted_upon_(self, a, self_on_left):
             ret = CombinatorialFreeModule.Element._acted_upon_(self, a, self_on_left)
             if ret is not None:  # did the scalar action
                 return ret
-            if self_on_left: # i.e., module element on left
+            if self_on_left:  # i.e., module element on left
                 a = a.antipode()
             b = a.change_basis('adem')
             ans = self.parent().zero()
@@ -1283,7 +1283,7 @@ def steenrod_module_map(self, deg_domain, deg_codomain, side='left'):
           the action as a left module action or a right module
 
         We will write this with respect to the left action;
-        for the right action, just switch all of the the tensors.
+        for the right action, just switch all of the tensors.
         Writing `m` for ``deg_domain`` and `n` for ``deg_codomain``, this
         returns `A^{n-m} \otimes H^{m} \to H^{n}`, one single
         component of the map making `H` into an `A`-module.

src/sage/rings/complex_interval.pyx

@@ -2237,7 +2237,7 @@ cdef _circle_invert_standard(
     # Consider the images
     #          f(xmin + ymin * I), ..., f(xmax + ymax * I)
     # of the four corners of the input rect under inversion f.
-    # Now consider the the axis-parallel rectangle R that these images span.
+    # Now consider the axis-parallel rectangle R that these images span.
     # In general, the image of the input rect might not be contained in R.
     # In case 1, however, (and only in case 1) it is and we furthermore know
     # which image is mapped to which edge of R. Thus, we have:

src/sage/rings/polynomial/polynomial_element.pyx

@@ -2280,7 +2280,7 @@ cdef class Polynomial(CommutativePolynomial):
 
         - ``degree`` -- ``None`` or positive integer (default: ``None``).
           Used for polynomials over finite fields. If ``None``, returns
-          the the first factor found (usually the smallest). Otherwise,
+          the first factor found (usually the smallest). Otherwise,
           attempts to return an irreducible factor of ``self`` of chosen
           degree ``degree``.
 

src/sage/schemes/elliptic_curves/ell_number_field.py

@@ -290,7 +290,8 @@ def simon_two_descent(self, verbose=0, lim1=2, lim3=4, limtriv=2,
         # time (when known_points may have increased) will not cause
         # another execution of simon_two_descent.
         try:
-            result = self._simon_two_descent_data[lim1,lim3,limtriv,maxprob,limbigprime]
+            result = self._simon_two_descent_data[lim1, lim3, limtriv,
+                                                  maxprob, limbigprime]
             if verbose == 0:
                 return result
         except AttributeError:
@@ -2343,7 +2344,7 @@ def gens(self, **kwds):
             sage: gg=E.gens(lim3=13); gg  # long time (about 4s)
             [(... : 1)]
 
-        Check that the the point found has infinite order, and that it is on the curve::
+        Check that the point found has infinite order, and that it is on the curve::
 
             sage: P=gg[0]; P.order()  # long time
             +Infinity
@@ -2447,7 +2448,7 @@ def period_lattice(self, embedding):
              -0.14934463314391922099120107422 - 2.0661954627294548995621225062*I)
         """
         from sage.schemes.elliptic_curves.period_lattice import PeriodLattice_ell
-        return PeriodLattice_ell(self,embedding)
+        return PeriodLattice_ell(self, embedding)
 
     def real_components(self, embedding):
         """