Bolzano–Weierstrass theorem

The theorem of Bolzano - Weierstrass ( by Bernard Bolzano and Karl Weierstrass ) is a set of analysis.

• 3.1 finite-dimensional vector spaces
• 3.2 infinite-dimensional vector spaces

Statement

First version

Every bounded sequence of complex numbers ( with infinitely many elements ) contains (at least) a convergent subsequence.

Second version

• Every bounded sequence of complex numbers ( with infinitely many members ) has ( at least) one accumulation point.
• Every bounded sequence of real numbers has a largest and a smallest accumulation point.

Sketch of proof

Evidence of general statements is returned to the one-dimensional real message. This one can prove, by simultaneously constructs a nest of intervals and a subsequence such that for each. These two sequences are recursively constructed.

To determine the largest accumulation point, one must, whenever possible, choose the upper subinterval, for the smallest accumulation point of the lower sub-interval.

Generalizations

Finite dimensional vector spaces

The complex numbers are viewed in the context of this sentence as a two-dimensional real vector space. For a sequence of column vectors with n real component, first select a sub-sequence which converges in the first component. This is chosen again a partial sequence that converges into the second component. The convergence in the first component is maintained since subsequences of convergent sequences are convergent with the same limit again. And so on until the n-th subsequence is converged in the last component.

Infinite-dimensional vector spaces

The theorem of Bolzano - Weierstrass is not valid in infinite-dimensional normed vector spaces. Thus, for example, the sequence of unit vectors (0,0, ..., 0,1,0, ..., 0, ... ) is limited in the sequence space, but does not limit point, as all sequence elements at a distance of from each other. This counter-example can be generalized to any infinite-dimensional normed spaces, one can always construct is an infinite sequence of vectors of length 1, with each pair have a distance of at least 1/2.

As a replacement for the theorem of Bolzano - Weierstrass in infinite-dimensional vector spaces exist in reflexive spaces following statement: Every bounded sequence of a reflexive space has a weakly convergent subsequence. Often together with the Sobolev embedding sets provides the existence of weakly convergent subsequences limited impact solutions of variational problems and thus partial differential equations.

Inferences and generalizations

It follows from the theorem of Bolzano - Weierstrass that every monotone and bounded sequence of real numbers converges ( monotonicity criterion) and that a continuous function on a closed and bounded interval a maximum and assumes a minimum ( set by the minimum and maximum).

The theorem of Bolzano - Weierstrass is closely related to the theorem of Heine- Borel. A generalization of both sets to topological spaces is the following: A topological space is then a compact space if every net has a convergent subnet.