The Kakutani fixed point theorem applies to functions operating on subsets of this. The standard
 formulation of the Vitali covering lemma has a collection in this as its subject. Transformations in these are
 called rigid motions. One way to generalize length to subsets of this is the Lebesgue measure. Each point in
 a manifold has a neighborhood that is homeomorphic to a set in one of these. The linear combinations of
 either the rows or columns of a real valued matrix make a closed, zero containing subset of this. It is a
 metric space whose metric function is the Pythagorean theorem for its dimension and its inner product is
 the dot product. For 10 points, name these spaces symbolized by a blackboard capital R to the n, which are
 name
The Kakutani fixed point theorem applies to functions operating on subsets of this. The standard
 formulation of the Vitali covering lemma has a collection in this as its subject. Transformations in these are
 called rigid motions. One way to generalize length to subsets of this is the Lebesgue measure. Each point in
 a manifold has a neighborhood that is homeomorphic to a set in one of these. The linear combinations of
 either the rows or columns of a real valued matrix make a closed, zero containing subset of this. It is a
 metric space whose metric function is the Pythagorean theorem for its dimension and its inner product is
 the dot product. For 10 points, name these spaces symbolized by a blackboard capital R to the n, which are
 named for the Greek author of the Elements.
The Kakutani fixed point theorem applies to functions operating on subsets of this.
{Euclidean} Space [prompt on {metric space}]