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In simple economic models, it is common to assume that there is only a finite number of different commodities, e.g. houses, fruits, cars, etc., so every bundle can be represented by a finite vector, and the consumption set is a vector space with a finite dimension. But in reality, the number of different commodities may be infinite.
The set of one-dimensional subspaces of a fixed finite-dimensional vector space V is known as projective space; it may be used to formalize the idea of parallel lines intersecting at infinity. [98] Grassmannians and flag manifolds generalize this by parametrizing linear subspaces of fixed dimension k and flags of subspaces, respectively.
The dual of ℓ ∞ is the ba space. The spaces c 0 and ℓ p (for 1 ≤ p < ∞) have a canonical unconditional Schauder basis {e i | i = 1, 2,...}, where e i is the sequence which is zero but for a 1 in the i th entry. The space ℓ 1 has the Schur property: In ℓ 1, any sequence that is weakly convergent is also strongly convergent .
An infinite-dimensional vector function is a function whose values lie in an infinite-dimensional topological vector space, such as a Hilbert space or a Banach space. Such functions are applied in most sciences including physics .
Both vector addition and scalar multiplication are trivial. A basis for this vector space is the empty set, so that {0} is the 0-dimensional vector space over F. Every vector space over F contains a subspace isomorphic to this one. The zero vector space is conceptually different from the null space of a linear operator L, which is the kernel of L.
In mathematics, the L p spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces.They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz ().
The space ℓ ∞ is not separable, and therefore has no Schauder basis. Every orthonormal basis in a separable Hilbert space is a Schauder basis. Every countable orthonormal basis is equivalent to the standard unit vector basis in ℓ 2. The Haar system is an example of a basis for L p ([0, 1]), when 1 ≤ p < ∞. [2]
However, this geometric equivalence between L 1 and L ∞ metrics does not generalize to higher dimensions. A sphere formed using the Chebyshev distance as a metric is a cube with each face perpendicular to one of the coordinate axes, but a sphere formed using Manhattan distance is an octahedron : these are dual polyhedra , but among cubes ...