Question

Let \(C\) be a closed convex bounded subset of a Banach space \(X\). Show that if \(T: C \rightarrow C\) is a nonexpansive map, then \(\inf \\{\|x-T(x)\| ; x \in C\\}=0\).

Short answer

\(\inf \{\|x - T(x)\| ; x \in C\} = 0\).

Step-by-step solution

- Understand the Given Information

We are given a closed convex bounded subset \(C\) of a Banach space \(X\). There is a nonexpansive map \(T: C \rightarrow C\), meaning for all \(x, y\) in \(C\), we have \(\|T(x) - T(y)\| \leq \|x - y\|\). We need to show that \(\inf \{\|x - T(x)\| ; x \in C\} = 0\).

- Define the Fixed Point Condition

Consider that nonexpansive maps on convex bounded subsets of Banach spaces have fixed points or the property we're investigating. We need to demonstrate that the distance \(\|x - T(x)\|\) can get arbitrarily small.

- Utilize the Banach Fixed-Point Theorem (if necessary)

Although \(T\) is nonexpansive and not necessarily a contraction, cheaper strategies might fail here, so leveraging Banach's theorem is plausible, but our approach should seek simplicity. Hence, we inspect \(T\)'s iterates.

- Consider the Sequence of Iterates

Define a sequence \(x_n\) by selecting \(x_0 \in C\) and setting \(x_{n+1} = T(x_n)\). Since \(C\) is bounded and \(T\) is nonexpansive, this sequence \(\{x_n\}\) remains within \(C\) and follows \(\|x_{n+1} - x_{n+2}\| = \|T(x_n) - T(x_{n+1})\| \leq \|x_n - x_{n+1}\|\).

- Show Convergence of Distances

Because \(x_{n+1} - x_n = x_n - T(x_n)\), nonexpansiveness induces a Cauchy sequence property through \(\|x_{n+1} - x_{n+2}\| \leq \|x_n - x_{n+1}\| \rightarrow 0\). Hence, the consequence \(\|x_n - T(x_n)\| \rightarrow 0\) follows.

- Establish the Infimum Result

Given \(\|x_n - T(x_n)\| \rightarrow 0\), every convergent subsequential limit within \(C\) will force the infimum of \(\|x - T(x)\|\) over \(x \in C\) to be zero.

Key concepts

Banach Space

A Banach Space is a type of vector space that is important in the field of functional analysis. It is complete, meaning every Cauchy sequence in the space converges to a point within the space. This completeness is crucial because it ensures that limits of sequences behave in a predictable way.

Key properties of a Banach Space are:

• It is a vector space over the field of real or complex numbers.
• It is equipped with a norm \( \|\bullet \| \), which measures the 'size' of its elements.
• It is complete with respect to the norm-induced metric \( d(x, y) = \|x - y\| \).

The concept of Banach Space is essential for understanding various fixed-point theorems and methods for solving differential equations.

Closed Convex Bounded Set

A Closed Convex Bounded Set has specific properties that make it a fundamental concept in analysis:

• **Closed**: The set contains all its limit points. If a sequence within the set converges, its limit also belongs to the set.
• **Convex**: If you take any two points within the set and draw a line between them, all the points on that line are also inside the set.
• **Bounded**: All points in the set are within some fixed distance from each other.

These properties ensure the set is well-behaved in terms of optimization and convergence. When dealing with nonexpansive maps, working within such sets guarantees that iterative methods can be applied effectively.

Fixed Point Theorem

A Fixed Point Theorem states that under certain conditions, a function will have at least one fixed point. That is, a point \( x \) such that \( T(x) = x \):

• The **Banach Fixed-Point Theorem** applies to contraction mappings in complete metric spaces, asserting the existence and uniqueness of a fixed point.
• For **nonexpansive maps**, which do not necessarily contract distances, fixed-point results can still be derived under suitable conditions, such as in closed convex bounded subsets of Banach spaces.

Fixed-point theorems are critical for proving the existence of solutions in various mathematical and real-world problems where direct solutions are challenging to identify.

Cauchy Sequence

A Cauchy Sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. Formally, a sequence \( {x_n} \) is Cauchy if for every \( \epsilon > 0 \, \) there exists an integer \( N \) such that for all \( m, n > N \), \|x_m - x_n\| < \epsilon \. This concept is vital for understanding convergence in metric spaces.
In a Banach space, every Cauchy sequence converges to a point within the space, which is a crucial property used in fixed-point theory and in proving that nonexpansive maps bring elements as close as desired.