* Take the Banach space R < sup > n </ sup > ( or C < sup > n </ sup >) with norm || x ||

The spectrum of any element x is a closed subset of the closed disc in C with radius || x || and center 0, and thus is compact.

* || x *||

|| x ||< sup > 2 </ sup > for all x in A.

* If V is a normed vector space with linear subspace U ( not necessarily closed ) and if z is an element of V not in the closure of U, then there exists a continuous linear map with ψ ( x ) = 0 for all x in U, ψ ( z ) = 1, and || ψ || = 1 / dist ( z, U ).

The sphere is the inverse image of a one-point set under the continuous function || x ||.

If X and Y are subsets of the real numbers, d < sub > 1 </ sub > and d < sub > 2 </ sub > can be the standard Euclidean norm, || · ||, yielding the definition: for all ε > 0 there exists a δ > 0 such that for all x, y ∈ X, | x − y | < δ implies | f ( x ) − f ( y )| < ε.

*|| Tx || = || T * x || for all x ( use ).

| 1958 – 59 || 39 || 0 || 2 || 0 || 2 || 0 || 44 *|| 0

| 1967 – 68 || 34 || 5 || 1 || 0 || 10 || 0 || 2 || 0 || 50 *|| 6 †

486 || 22 || 94 || 7 || 137 || 4 || 69 || 3 || 789 *|| 37 †

* In particular, if V is a normed vector space and if z is any element of V, then there exists a continuous linear map with ψ ( z ) = || z || and || ψ || ≤ 1.

| align = left | Doe Maar || 1979 ||-||||||

| align = left | Skunk || 1981 || 27-06-1981 || 1 ( 3wk )|| 56 ||

| align = left | Doe de dub-Discodubversie || 1982 ||-||||||

| align = left | Doris Day en andere stukken || 1982 || 27-03-1982 || 1 ( 3wk )|| 59 ||

| align = left | 4us ( Virus )|| 1983 || 26-03-1983 || 1 ( 5wk )|| 21 ||

| align = left | Lijf aan lijf || 1983 || 12-11-1983 || 22 || 14 || Live album

As an example, the field of real numbers is not algebraically closed, because the polynomial equation x < sup > 2 </ sup > + 1 = 0 has no solution in real numbers, even though all its coefficients ( 1 and 0 ) are real.

Since p ( x ) is irreducible, this means that p ( x ) = k ( x − a ), for some k ∈ F

In more general sense, each column is treated as a polynomial over GF ( 2 < sup > 8 </ sup >) and is then multiplied modulo x < sup > 4 </ sup >+ 1 with a fixed polynomial c ( x ) = 0x03 · x < sup > 3 </ sup > + x < sup > 2 </ sup > + x + 0x02.

* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

For example, the equation y = x corresponds to the set of all the points on the plane whose x-coordinate and y-coordinate are equal.

These points form a line, and y = x is said to be the equation for this line.

This is not always the case: the trivial equation x = x specifies the entire plane, and the equation x < sup > 2 </ sup > + y < sup > 2 </ sup > = 0 specifies only the single point ( 0, 0 ).

The equation x < sup > 2 </ sup > + y < sup > 2 </ sup > = r < sup > 2 </ sup > is the equation for any circle with a radius of r.

For example, the parent function y = 1 / x has a horizontal and a vertical asymptote, and occupies the first and third quadrant, and all of its transformed forms have one horizontal and vertical asymptote, and occupies either the 1st and 3rd or 2nd and 4th quadrant.

In general, if y = f ( x ), then it can be transformed into y = af ( b ( x − k )) + h. In the new transformed function, a is the factor that vertically stretches the function if it is greater than 1 or vertically compresses the function if it is less than 1, and for negative a values, the function is reflected in the x-axis.