Mathematics is a game played according to certain simple rules and meaningless marks on paper.

—DAVID HILBERT (1862–1943)

When you try to prove a theorem, you don’t just list the hypotheses, and start to reason. What you do is trial and error, experimentation, guesswork.

—PAUL R. HALMOS (1916–2006)

Vector Spaces

There are many vector spaces other than the spaces that we have been using heretofore. What type of mathematical constructs will be called “vector spaces”? The structure they must have is set out in the following formal definition.


A vector space is a set V of elements (called vectors), together with ...

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