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A sphere in 3-space (also called a 2-sphere because it is a 2-dimensional object) consists of the set of all points in 3-space at a fixed distance r from a central point P. The solid enclosed by the sphere is called a ball (or, more precisely a 3-ball). The volume of the ball is given by
Consider the space of all functions from X to F which vanish on all but a finite number of points in X. This space is a vector subspace of F X, the space of all possible functions from X to F. To see this, note that the union of two finite sets is finite, so that the sum of two functions in this space will still vanish outside a finite set.
A three-dimensional Euclidean space is a special case of a Euclidean space. In Bourbaki's terms, [2] the species of three-dimensional Euclidean space is richer than the species of Euclidean space. Likewise, the species of compact topological space is richer than the species of topological space. Fig. 3: Example relations between species of spaces
Let the field K be the set R of real numbers, and let the vector space V be the Euclidean space R 3. Consider the vectors e 1 = (1,0,0), e 2 = (0,1,0) and e 3 = (0,0,1). Then any vector in R 3 is a linear combination of e 1, e 2, and e 3. To see that this is so, take an arbitrary vector (a 1,a 2,a 3) in R 3, and write:
This may be seen by differentiating the orthogonality condition, A T A = I, A ∈ SO(3). [nb 2] The Lie bracket of two elements of () is, as for the Lie algebra of every matrix group, given by the matrix commutator, [A 1, A 2] = A 1 A 2 − A 2 A 1, which is again a skew-symmetric matrix
The subset of the integers {0,1,2} is contained in the interval of real numbers [0,2], which is convex. The Shapley–Folkman lemma implies that every point in [0,2] is the sum of an integer from {0,1} and a real number from [0,1]. [7] The distance between the convex interval [0,2] and the non-convex set {0,1,2} equals one-half:
where f (2k−1) is the (2k − 1)th derivative of f and B 2k is the (2k)th Bernoulli number: B 2 = 1 / 6 , B 4 = − + 1 / 30 , and so on. Setting f ( x ) = x , the first derivative of f is 1, and every other term vanishes, so [ 15 ]
If V is a vector space over a field K, a subset W of V is a linear subspace of V if it is a vector space over K for the operations of V.Equivalently, a linear subspace of V is a nonempty subset W such that, whenever w 1, w 2 are elements of W and α, β are elements of K, it follows that αw 1 + βw 2 is in W.