# What is an honest basis?

In a comment to this question, the commentator stated that

"the monomials form an honest basis for your vector space".

To be honest, I never heard of that. Is this something elementary?

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I mean ‘honest’ in the usual sense of the English word. There are other kinds of bases for special vector spaces: orthogonal bases, Schauder bases... – Zhen Lin Jan 17 '12 at 14:10
Thank you for your honesty. – Paul Jan 17 '12 at 14:15
Enough puns, honestly! – joriki Jan 17 '12 at 14:51
One standard term for an "honest basis", as opposed to a Schauder basis, is "Hamel basis". – Brad Jan 18 '12 at 0:51

This was not a term of art, but rather an attempt at emphasis. The monomials form a basis for the vector space in the usual sense (every vector is a linear combination of elements of the basis in a unique way), as opposed to, for instance, a Hilbert basis (whose linear span is not necessarily equal to the entire space).

(Reminds me of something that happened when I was taking Measure Theory in my final undergraduate year; the professor had his own very good notes, with a set of exercises. One of the problems asked us to prove that a function that satisfied a certain property "is automatically continuous"; we couldn't figure out what the definition of "automatically continuous" was, and asked the professor the next lecture. Of course, he meant that such a function would necessarily be continuous...)

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