## How do you show that a complex function is bounded?

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If f is real-valued and f(x) ≤ A for all x in X, then the function is said to be bounded (from) above by A. If f(x) ≥ B for all x in X, then the function is said to be bounded (from) below by B.

**How do you prove Weierstrass Theorem?**

The first proof of the Weierstrass Theorem makes use of Bernstein polynomials. This method of proof gives a constructive method of finding a sequence of polynomials which converge uniformly on the interval to the given continuous function. Also, the rapid ity of the convergence can be estimated.

### Why is the Weierstrass approximation theorem important?

In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly approximated as closely as desired by a polynomial function.

**Is C1 dense in c0?**

C1[[0,1] contains the set of all polynomials restricted to [0,1], which is dense in C[0,1].

## What is Liouville’s theorem in complex analysis?

In complex analysis, Liouville’s Theorem states that a bounded holomorphic function on the entire complex plane must be constant. It is named after Joseph Liouville.

**What is the integration technique called weierstrass substitution?**

The Weierstrass substitution, named after German mathematician Karl Weierstrass (1815−1897), is used for converting rational expressions of trigonometric functions into algebraic rational functions, which may be easier to integrate.

### What is Weierstrass a theorem?

A theorem obtained and originally formulated by K. Weierstrass in 1860 as a preparation lemma, used in the proofs of the existence and analytic nature of the implicit function of a complex variable defined by an equation $ f ( z, w) = 0 $ whose left-hand side is a holomorphic function of two complex variables.

**What is Weierstrass’theorem on the approximation of functions?**

Weierstrass’ theorem on the approximation of functions: For any continuous real-valued function $ f ( x) $ on the interval $ [ a, b] $ there exists a sequence of algebraic polynomials $ P _ {0} ( x), P _ {1} ( x) \\dots $ which converges uniformly on $ [ a, b] $ to the function $ f ( x) $; established by K. Weierstrass .

## Can Weierstrass’infinite product theorem be generalized to an arbitrary domain?

Weierstrass’ infinite product theorem can be generalized to the case of an arbitrary domain $ D \\subset \\mathbf C $: Whatever a sequence of points $ \\ { \\alpha _ {k} \\} \\subset D $ without limit points in $ D $, there exists a holomorphic function $ f $ in $ D $ with zeros at the points $ \\alpha _ {k} $ and only at these points.

**Is the Stone-Weierstrass theorem valid for all real-valued functions?**

The theorem is also valid for real-valued continuous $ 2 \\pi $- periodic functions and trigonometric polynomials, e.g. for real-valued functions which are continuous on a bounded closed domain in an $ m $- dimensional space, or for polynomials in $ m $ variables. For generalizations, see Stone–Weierstrass theorem.