Question
A vector space is Banach if it is complete with respect to this function. For 10 points each:
[10m] Name this kind of function that generalizes a vector's length. A metric can be defined by applying this function to the difference of two vectors.
ANSWER: norms [prompt on absolute value]
[10e] If a vector space has two nonequivalent norms, its dimension must have this property. If a vector space's dimension has this property, it is isomorphic to a proper subspace. That is analogous to the fact that a set can be in bijection with a proper subset if it has this property, which Hilbert illustrated with paradox involving a hotel with this property.
ANSWER: infinite [or word forms such as infinity]
[10h] In infinite-dimensional spaces, the closed unit ball lacks this property. The Brouwer fixed point theorem applies to continuous mappings from a nonempty convex set with this property to itself.
ANSWER: compactness
<Andrew Rout , Science - Other - Math Pure>
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