Question

When this quantity is infinite, an object named for Hamel (“HAHM-uhl”) may be replaced by one named for Schauder (“SHAO-der”). This quantity for the degree-k exterior algebra of X equals this quantity for X choose k. Two values for this quantity, rank and nullity, (20[2])sum to this quantity (20[3])for the (20[1])domain (20[1])of a linear transformation. (20[1])The (20[1])(*) “Minkowski–Bouligand” (10[1])(“min-COW-ski boo-lee-GAHND”) form of this quantity (10[1])is alternatively named (10[1])for counting (10[1])boxes. (10[1])This quantity equals the cardinality of a basis set. (10[1])This (10[1])quantity’s “Hausdorff” (10[1])form (10[6])describes the coastline (10[1])of England’s (10[1])“fractional” value of this (10[1])quantity, which characterizes fractals. This quantity equals the number of coordinates in Euclidean space. (10[2])For 10 (10[1])points, name this integer geometric quantity denoted d. (10[1])■END■ (10[3])

ANSWER: dimension [accept Minkowski–Bouligand dimension or Minkowski dimension or box-counting dimension or fractional dimension or fractal dimension; prompt on cardinality of a basis set before “cardinality” is read by asking “what quantity is defined as the cardinality of a basis set?”] (For a vector space with uncountably infinite dimension, a basis can be referred to as a “Hamel basis,” in contrast to Schauder bases, which do not satisfy the standard requirements for a basis, but are countable.)
<David Bass, Science - Math&gt; ~27909~ &lt;Editor: David Bass>
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