basis

noun8

1. •
   something that an argument or a theory starts from and depends on

   • I wonder if the South Korean side has any basis that its smog is from China. — 2019
2. •
   A regular frequency.

   • You should brush your teeth on a daily basis at minimum.
   • The flights to Fiji leave on a weekly basis.
3. •
   A physical base or foundation.

   • Beholding rocks from their firm basis rent; Mountain on mountain thrown, With threatening hurl, that shook th' aerial firmament! — 1695
4. •
   An underlying condition or circumstance.

   • Allow us to compete on an equal basis.
5. •
   In a vector space, a linearly independent set of vectors spanning the whole vector space.
6. •
   Amount paid for an investment, including commissions and other expenses.
7. •
   The difference between the cash price a dealer pays to a farmer for his produce and an agreed reference price, which is usually the futures price at which the given crop is trading at a commodity exchange.

   • Included in the basis could be elevation, cleaning, freight by truck and/or rail, government inspection fees, administration fees, interest and storage charges as well as allowance for risk and profit for the grain dealer.
8. •
   A collection of subsets ("basis elements") of a set, such that this collection covers the set, and for any two basis elements which both contain an element of the set, there is a third basis element contained in the intersection of the first two, which also contains that element.

   • The collection of all possible unions of basis elements of a basis is said to be the topology generated by that basis.