Hilbert Space An inner product space which is
complete with respect to the norm induced by the inner product.
Examples: L2 (modulo almost everywhere equality), l2, Euclidean space
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Banach space A complete, normed, linear space
Examples: Lp spaces, continuous functions on a compact set under the
supremum norm
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Inner product space A vector space which is endowed
with an inner product.
Examples: The real numbers with
multiplication, Euclidean space with the dot product
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Vector space A set of vectors together with a field of
scalars and two operations: vector addition and scalar
multiplication. Vector addition is associative and commutative, and
the two operations distribute in a natural way.
Examples:
Euclidean space, m by n matrices
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Polish space A complete, separable, metric space.
Examples: Euclidean space, many probability spaces
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Measure space A triple consisting of a set, a
sigma-algebra of subsets of the set, and a measure compatible with
that sigma-algebra.
Examples: any probability space, the Real
numbers under Lebesgue measure
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Outer space The vast region of the universe not
occupied by planets, stars, or other heavenly bodies.
Example: The region between the moon and New York City
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Sobolev space A vector space of functions equipped
with a norm which depends on derivatives of the given function. The
derivatives are understood in a weak sense, so that the space is
complete (so it is a Banach space).
Examples: W(k,p)
for various k and p .
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Probability space A measure space where the measure,
typically called P , is countably additive and has total mass
1.
Examples: The real numbers, equipped with the measures
induced by various probability distributions.
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My space A popular social networking website.
Example: myspace.com
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Topological space A set, together with a collection of
subsets which are to be the open sets, satisfying certain properties.
Examples: Euclidean space, any manifold, an algebraic variety
with the Zariski topology
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Office space
A popular 1999 movie staring Ron Livingston, Jennifer Aniston, and
Stephen Root
Examples: Red Swingline stapler
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