Let be a real vector space (e.g., the real continuous functions on a closed interval , two-dimensional Euclidean space , the twice differentiable real functions on , etc.). Then is a real subspace of if is a subset of and, for every , and (the reals), and . Let be a homogeneous system of linear equations in , ..., . Then the subset of which consists of all solutions of the system is a subspace of .
More generally, let be a field with , where is prime, and let denote the -dimensional vector space over . The number of -D linear subspaces of is
(1)
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where this is the q-binomial coefficient (Aigner 1979, Exton 1983). The asymptotic limit is
(2)
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where
(3)
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(4)
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(5)
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(6)
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(Finch 2003), where is a Jacobi theta function and is a q-Pochhammer symbol. The case gives the q-analog of the Wallis formula.