Question

(Korovkin) Let \(\left\\{T_{n}\right\\}\) be a sequence of bounded linear operators on \(C[0,1]\) that are positive; that is, \(T_{n}(f) \geq 0\) if \(f \geq 0\) on \([0,1]\). Assume that \(T_{n}(1) \rightarrow 1, T_{n}(x) \rightarrow x\), and \(T_{n}\left(x^{2}\right) \rightarrow x^{2}\) in \(C[0,1]\). Show that \(T_{n}(f) \rightarrow f\) in \(C[0,1]\) for every \(f \in C[0,1]\). Is it true that \(\left\|T_{n}-I_{C[0,1]}\right\| \rightarrow 0\) ?

Short answer

By Korovkin's theorem, \({T_{n}(f) \rightarrow f}\) holds for all \({f \in C[0,1]}\), and \({\|T_{n} - I_{C[0,1]}\| \rightarrow 0}\).

Step-by-step solution

Understanding the Assumptions and Conditions

We are given a sequence of bounded linear operators \({T_{n}}\) on \({C[0,1]}\) that are positive. This means \({T_{n}(f) \geq 0\)}\ if \({f \geq 0\)}\ on \({[0,1]}\). The conditions state that \({T_{n}(1) \rightarrow 1}\), \({T_{n}(x) \rightarrow x}\), and \({T_{n}(x^2) \rightarrow x^2}\) in \({C[0,1]}\). We need to show that \({T_{n}(f) \rightarrow f}\) in \({C[0,1]}\) for every \({f \in C[0,1]}\).

Using Korovkin's Theorem

Korovkin's theorem provides conditions under which convergence to the identity operator can be guaranteed. According to the theorem, if \({T_{n}(1) \rightarrow 1}\), \({T_{n}(x) \rightarrow x}\), and \({T_{n}(x^2) \rightarrow x^2}\) uniformly in \({C[0,1]}\), then \({T_{n}(f) \rightarrow f}\) uniformly for all \({f \in C[0,1]}\).

Applying the Conditions

Given that \({T_{n}(1) \rightarrow 1}\), \({T_{n}(x) \rightarrow x}\), and \({T_{n}(x^2) \rightarrow x^2}\) in \({C[0,1]}\) by assumption, these match exactly with the criteria specified by Korovkin's theorem. Therefore, \({T_{n}(f) \rightarrow f}\) uniformly in \({C[0,1]}\) for every \({f \in C[0,1]}\).

Conclusion on Uniform Norm Convergence

Since we established that \({T_{n}(f) \rightarrow f}\) uniformly in \({C[0,1]}\), this implies that the norm \({\|T_{n} - I_{C[0,1]}\| \rightarrow 0}\). This is because uniform convergence of operators on a normed space directly translates to norm convergence.

Key concepts

Bounded Linear Operators

Bounded linear operators are fundamental in functional analysis. They map functions from one space to another while maintaining linearity and ensuring that output values do not exponentially increase. This means there exists a constant \(C\) such that for any function \(f\) and its image under \(T\), we keep \[\text{||} T(f) \text{||} \leq C \text{||} f \text{||}.\]

The linearity property indicates that \[ T(f + g) = T(f) + T(g) \text{ and } T(\text{a}f) = \text{a}T(f).\]
These transformations are straightforward once we understand their restrictions and outputs, aligning perfectly with the given function's properties. In the provided problem, \ \text{T}_n\ are bounded, meaning they comfortably fit the criteria, providing us confidence within the bounds.

Uniform Convergence

Uniform convergence is when we talk about a sequence of functions converging to a specific function uniformly across their entire domain. This entails no point in the domain experiencing a significantly different rate or degree of convergence. Formally, \ T_n \ tends to \ T \uniformly if for any given \ \text{ε } > 0 \, \ exists an N such that for every \ \text{n} \
if \ n \, n > N \ implies \[ \text{||} T_n(x) - T(x) \text{||} < \text{ε} \text{for all} x \text{ in } X.\]
For Korovkin's Theorem, this is key because uniform convergence of \T_n \ to identity operator guarantees that for every function within \ C[0,1] \, the sequence will converge uniformly to the function itself without exception.

Positive Operators

Positive operators are those which, when acting on a non-negative function, yield another non-negative function. In simpler words, if a function \[ f(x) \geq 0 \] for every \ x \ in range, then \ T_n(f) \ too will hold \geq 0 \ for every \ x \ in range \ [0, 1].\

This comes handy in understanding Korovkin's theorem in our context because ensuring positivity means ensuring the gradual step-wise approximation \ T_n \ for each function won’t result in any unexpected negative results. Maintaining this property is crucial while shifting our analysis.

Norm Convergence

Norm convergence deals with functions and their approximating sequences converging per their norms, which determines 'distance' in their specific space. It is dictated by
\[ ||T_n(f) - f|| \rightarrow 0 \text{ as } n \rightarrow \text{∞}.\]
This denotes how close we are to the original function in terms of our norm.
For Korovkin's theorem, this kind of convergence tells us directly about how our bounded linear sequences become the identity operator, complying neatly with topic and problem constraints. Norm convergence further solidifies the conclusion that \ T_n(f) \ approximates appropriately across the entire function space.