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A norm on a vector space is:
A mapping from elements of the vector space to the real numbers; it is positive for every element of the space, and zero only for the zero element. The norm of element v is denoted - - v - -
If n is a real number, then - - nv - - = - n - . - - v - -
|| u+v || ≤ || u || + || v || , for all u and v in the space.
A norm may be used to define a metric: d(x,y)= - - x-y - - .
A norm on a matrix is a norm on the vector space of matrices.

See normed vector space.
Of an element β of a finite field extension L/K : the quantity defined as follows. Choose a basis v₁, …,v_n of L over K , and define i( β) to be the linear map that takes v_i → βv_i . Then the norm

NL/K( β) = det i( β).

Also, the trace of β is

TrL/K( β) = Tr i( β).

The norm is multiplicative and the trace is additive in the usual way. The following formulae are also useful: for K ⊂ L ⊂ M we have
N_M/K=N_L/K °N_M/L,
Tr_M/K = Tr_L/K °Tr_M/L.