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94 Chapter Two. Vector Spaces 2.6 Example Another example of a subspace that is not a subset of an R n followed the definition of a vector space. The space in Example 1.12 of all real-valued functions of one real variable { f | f : R ! R } has the subspace in Example 1.14 of functions satisfying the restriction ( d 2 f/dx 2 ) + f = 0 . 2.7 Example The definition requires that the addition and scalar multiplication operations must be the ones inherited from the larger space. The set S = { 1 } is a subset of R 1 . And, under the operations 1 + 1 = 1 and r · 1 = 1 the set S is a vector space, specifically, a trivial space. However, S is not a subspace of R 1 because those aren’t the inherited operations, since of course R 1 has 1 + 1 = 2 . 2.8 Example Being vector spaces themselves, subspaces must satisfy the closure conditions. The set R + is not a subspace of the vector space R 1 because with the inherited operations it is not closed under scalar multiplication: if ~ v = 1 then - 1 · ~ v 62 R + . The next result says that Example 2.8 is prototypical. The only way that a subset can fail to be a subspace, if it is nonempty and uses the inherited operations, is if it isn’t closed.
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