nulsekvenser

Null sequence, or nulsekvens in Norwegian, refers to a sequence that tends to zero. In mathematics, a sequence (x_n) is called a null sequence if the limit of x_n as n grows without bound is 0. This concept is standard in real analysis and functional analysis, where it is used to describe terms that become arbitrarily small.

In the context of sequence spaces, the set of all real (or complex) null sequences is often denoted c0. With the sup norm, c0 forms a Banach space and sits inside the larger space of bounded sequences, l∞. A little-o notation related to null sequences is a_n = o(1), meaning a_n tends to zero faster than a fixed nonzero quantity.

Examples of null sequences include:

- The real sequence x_n = 1/n, which tends to 0.

- The sequence x_n = (-1)^n / n, which also tends to 0.

- The sequence x_n = 0 for all sufficiently large n, which trivially tends to 0.

Sequences that are not null include:

- x_n = sin(n), which does not converge to any limit, and in particular not to 0.

- x_n = 1 for all n, which converges to 1, not to 0.

- x_n = (-1)^n, which oscillates and has no limit.

Key properties include: if (a_n) → 0 and (b_n) is bounded, then (a_n b_n) → 0 in the real (or complex) setting. In normed spaces, a sequence (x_n) with ||x_n|| → 0 is called a null sequence, and c0 serves as an important example and tool in analysis.

Related concepts include little-o notation, null sequences in normed spaces, and the c0 space of sequences converging to zero.