Question

Let \(\mathcal{L}\) be a normed space of all Lipschitz functions on a Banach space \(X\) that are equal to 0 at the origin, under the norm $$ \|f\|=\sup \left\\{\frac{|f(x)-f(y)|}{\|x-y\|} ; x, y \in X\right\\} $$ Show that \(\mathcal{L}\) is a Banach space.

Short answer

The space \(\mathcal{L}\) is a Banach space because every Cauchy sequence in \(\mathcal{L}\) converges to a limit in \(\mathcal{L}\).

Step-by-step solution

Understanding the Norm

The given norm for the space \(\text{Lip}_0(X)\) is defined as \(\|f\(x\)\| = \sup \left\{ \frac{\|f(x) - f(y)\|}{\|x - y\|} ; x, y \in X \right\}\). This is known as the Lipschitz norm, which measures the maximum rate of change of the function \(f\).

Definition of Banach Space

To show that \(\text{Lip}_0(X)\) is a Banach space, we must prove that it is a complete space under the given norm. In other words, every Cauchy sequence in \(\text{Lip}_0(X)\) must converge to a limit that is also in \(\text{Lip}_0(X)\).

Consider a Cauchy Sequence

Let \(\{f_n\}\) be a Cauchy sequence in \(\text{Lip}_0(X)\). This implies that for every \(\epsilon > 0\), there exists an integer \(N\) such that \(\| f_n - f_m \| < \epsilon\) for all \(n, m \geq N\).

Properties of Cauchy Sequences

Because \(\{f_n\}\) is a Cauchy sequence, for any \(\epsilon > 0\), \(\| f_n(x) - f_m(x) \| < \epsilon\) for all \(n, m \geq N\) and for all \(x \in X\). This implies that \(\{f_n\}\) is pointwise convergent; let its pointwise limit be \(f\).

Show that the Limit is Lipschitz

We need to show that \(f\) is a Lipschitz function and \(f(0) = 0\). Note for any \(x, y \in X\): \(\|f_n(x) - f_n(y) \|\) is bounded above by \(L \|x-y\|\) where \(L\) is the Lipschitz constant of \(f_n\). Since \(\{f_n\}\) is Cauchy, we have \(\|f(x) - f(y)\| \leq L \|x - y\|\). Thus, \(f\) is Lipschitz.

Prove \(f(0) = 0\)

Since \(f_n(0) = 0\) for all \(n\), by pointwise convergence, \(f(0) = \lim_\{n \to \infty\} f_n(0) = 0\). Therefore, \(f \in \text{Lip}_0(X)\).

Convergence in the Lipschitz Norm

Finally, we need \(f_n \to f\) in the Lipschitz norm. For any \(\epsilon > 0\), there exists \(N\) such that for all \(n, m \geq N\), \(\|f_n - f_m\| < \epsilon\). Taking \(m \to \infty\), we get \(\|f_n - f\| \leq \epsilon\), showing that the sequence converges in the Lipschitz norm.

Conclusion: \(\mathcal{L}\) is Complete

Since every Cauchy sequence in \(\mathcal{L}\) converges to a limit in \(\mathcal{L}\), \(\mathcal{L}\) is a Banach space.

Key concepts

Lipschitz functions

Lipschitz functions are a special kind of function that have a bound on how rapidly they can change. If you have a function \(f: X \to \mathbb{R}\) defined on a space \(X\), it is called Lipschitz if there is a constant \(L\), known as the Lipschitz constant, such that for any two points \(x, y\) in \(X\), the inequality \(\| f(x) - f(y) \| \leq L \| x - y \|\) holds. This tells us that the function's change is not only limited but also proportionate to the distance between \(x\) and \(y\).

These functions ensure a certain 'smoothness' and they won't have drastic jumps. Understanding the concept of Lipschitz functions is important in various areas like numerical analysis, differential equations, and optimization.

Normed space

A normed space is a vector space equipped with a function called a 'norm', denoted usually by \(\| \, \|\). This norm assigns a length or size to each vector in the space. For a vector space \(V\), the norm \(\| v \|\) measures the 'magnitude' of the vector \(v\). This must satisfy the following properties:

• \( \| v \| \geq 0 \) (non-negativity)
• \( \| v \| = 0 \) if and only if \(v=0\) (identity of indiscernibles)
• \( \| a v \| = |a| \| v \| \) for scalar \(a\) in the field over which the vector space is defined (scalar multiplication)
• \( \| u + v \| \leq \| u \| + \| v \| \) (triangle inequality)

A Banach space is a specific type of normed space that is 'complete', meaning every Cauchy sequence in the space converges to a point within the space.

Cauchy sequence

A Cauchy sequence is a sequence of elements \(\{x_n\}\) in a metric space such that for every positive number \(\epsilon\), there is an integer \(N\) where the distance between any two terms \(x_n\) and \(x_m\) in the sequence is less than \(\epsilon\) for all \(n, m \geq N\). Mathematically, it is written as:
\[ \forall \epsilon > 0, \exists N \in \mathbb{N}, \forall n, m \geq N, \| x_n - x_m \| < \epsilon \]
This means that as you go further in the sequence, the terms get arbitrarily close to each other. The importance of Cauchy sequences lies in showing the completeness of a space. In a Banach space, every Cauchy sequence will converge to a limit that is also within that space.

Pointwise convergence

Pointwise convergence occurs when a sequence of functions \(\{f_n\}\) converges to a function \(f\) at each point individually in the domain. Formally, for each point \(x\) in the domain, the sequence of values \(\{f_n(x)\}\) converges to \(f(x)\):

\[ \forall x \in X, \; \lim_{n \to \infty} f_n(x) = f(x) \]

This type of convergence is different from uniform convergence, which requires the functions to get close to the limit function uniformly over the entire domain. Pointwise convergence is easier to meet, but doesn't give as strong guarantees about the behavior of the sequence of functions as a whole.