Weierstrass in a sentence

There are also more general versions of the Stone–Weierstrass that weaken the assumption of local compactness. citation Applications The Stone–Weierstrass theorem can be used to prove the following two statements which go beyond Weierstrass's result.

Application to economics There are different important equilibrium concepts in economics, the proofs of the existence of which often require variations of the Bolzano–Weierstrass theorem.

Existence The extreme value theorem of Karl Weierstrass states that a continuous real-valued function on a compact set attains its maximum and minimum value.

Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities, though the subject is still occasionally called "infinitesimal calculus".

His result is known as the Stone–Weierstrass theorem.

However, in 1872 Weierstrass found the first example of a function that is continuous everywhere but differentiable nowhere.

In 1881 he left for the University of Vienna to complete his mathematics studies under the supervision of Leo Königsberger (a former student of Weierstrass).

In fact, general topology tells us that a metrizable space is compact if and only if it is sequentially compact, so that the Bolzano–Weierstrass and Heine–Borel theorems are essentially the same.

In his work Weierstrass formalized the concept of limit and eliminated infinitesimals.

It can be proven by bisection method used in proving the Bolzano-Weierstrass theorem, the property (1) of ultrafilters turns out to be crucial.

It dealt with Weierstrass ' theorem on approximation to continuous functions by polynomials.

Now we may state: :Stone–Weierstrass Theorem (real numbers).

Particular types are Weierstrass's elliptic functions and Jacobi's elliptic functions and the sine lemniscate and cosine lemniscate functions.

Some fifty years later the result was identified as significant in its own right, and proved again by Weierstrass.

The Bolzano–Weierstrass theorem allows one to prove that if the set of allocations is compact and non-empty, then the system has a Pareto-efficient allocation.

The calculus of Leibniz and Newton was reexamined by mathematicians such as Weierstrass in order to rigorously prove the theorems thereof.

Then we may state : :Stone–Weierstrass Theorem (quaternion numbers).

These functions and their first derivative are related by the formula : Here, g 2 and g 3 are constants; is the Weierstrass elliptic function and its derivative.

The Stone–Weierstrass theorem mentioned above, for example, relies on Banach algebras which are both Banach spaces and algebras.

The subject has received later contributions at the hands of Weierstrass, Kronecker (Crelle, 101), and Méray.