Question

Let \(X_{k}\) be the space \(C^{k}[0,1]\) endowed with the norm $$ \|f\|=\sum_{j=1}^{k} \max \left\\{\left|f^{(j)}(t)\right| ; t \in[0,1]\right\\} $$ Show that the identity map from \(X_{k}\) into \(X_{k-1}\) is a compact operator.

Short answer

The identity map is compact by the Arzelà–Ascoli theorem, as bounded sets in \(X_k\) are relatively compact in \(X_{k-1}\).

Step-by-step solution

Understanding the Norms

First, recall the norms for the given spaces. For a function space \(C^{k}[0,1]\), the norm is defined as \(orm{f} = \sum_{j=1}^k \max \{ |f^{(j)}(t)| : t \in [0,1] \}.\) The space \(X_{k-1}\) has a similar norm but goes up to \(k-1\).

Identifying the Identity Map

The identity map here is the function \(I: X_k \to X_{k-1}\) where for any function \(f\in X_k\), \(I(f) = f\). We need to show that this map is compact; that is, it maps bounded sets in \(X_k\) to relatively compact sets in \(X_{k-1}\).

Relating Norms

Observe that \(orm{f}_{X_{k-1}} \leq orm{f}_{X_k}\) since the norm in \(X_k\) includes the norm in \(X_{k-1}\) plus additional terms. Hence, if a set is bounded in \(X_k\), it is also bounded in \(X_{k-1}\).

Arzelà–Ascoli Theorem

To show compactness, use the Arzelà–Ascoli theorem. A set of functions in \(C^{k-1}[0,1]\) is relatively compact if it is bounded and equicontinuous. Boundedness follows from the previous step. To show equicontinuity, note that derivatives up to \(k-2\) are bounded in \(X_{k-1}\).

Equicontinuity

For each bounded set in \(X_k\), the functions are differentiable, and the derivatives up to order \(k-1\) are bounded. Thus, these functions are equicontinuous on \([0,1]\) by the Arzelà–Ascoli theorem, implying compactness.

Conclusion

By the Arzelà–Ascoli theorem, the set of functions in \(X_{k-1}\) mapped from a bounded set in \(X_k\) is relatively compact. Thus, the identity map from \(X_k\) to \(X_{k-1}\) is compact.

Key concepts

normed spaces

Normed spaces are fundamental in functional analysis. They are vector spaces equipped with a norm, which is a function that assigns a length or size to each vector. In simpler terms, a norm gives us a way to measure how 'large' or 'small' vectors are. To formalize this, for any vector space \(V\), a norm is a function \(\| \cdot \|: V \to \mathbb{R}\) that must satisfy the following properties:

• Positivity: \(\|v\| \geq 0\) for all \(v \in V\) and \(\|v\| = 0\) if and only if \(v = 0\).
• Scalar multiplication: \(\|\alpha v\| = |\alpha| \cdot \|v\|\) for any scalar \(\alpha\) and any vector \(v \in V\).
• Triangle inequality: \(\|v + w\| \leq \|v\| + \|w\|\) for all \(v, w \in V\).

In the context of this exercise, we are dealing with spaces \(X_k\) and \(X_{k-1}\), which consist of functions and their derivatives up to a certain order, denoted by \(C^k[0, 1]\). The norm is given by the sum of the maximum magnitudes of these derivatives on the interval \([0, 1]\). This is written as:

\[\|f\| = \sum_{j=1}^k \max \{|f^{(j)}(t)| : t \in [0, 1]\}.\]

This specific norm tells us the “size” of a function within this space, considering the contributions from its derivatives up to order \(k\).

Arzelà–Ascoli theorem

The Arzelà–Ascoli theorem is a key result in the theory of function spaces. It provides a criterion for compactness of a set of functions in the space of continuous functions. To be more specific, a set \(S\) of functions within \(C[0, 1]\) (the space of continuous functions on \([0, 1]\)) is relatively compact if:

• It is bounded: There exists a constant \(M\) such that \(\|f\| \leq M\) for all \(f \in S\).
• It is equicontinuous: For every \(\epsilon > 0\), there exists a \(\delta > 0\) such that for all functions \(f \in S\) and for all pairs of points \(x, y \in [0, 1]\) with \(|x - y| < \delta\), \(|f(x) - f(y)| < \epsilon\).

In simpler words, not only must the functions in the set be uniformly bounded in magnitude, but they must also not oscillate too wildly; changes in their values should be controlled uniformly across the set. In this problem, we used the Arzelà–Ascoli theorem to show that the identity map is compact. By demonstrating boundedness from the norm properties and equicontinuity using derivatives, we applied this theorem to conclude that the set of functions in \(X_{k-1}\) is relatively compact.

function spaces

Function spaces are sets of functions with additional structure, such as algebraic or topological properties, which allow us to analyze functions as objects in their own right. In this exercise, the spaces \(X_k\) and \(X_{k-1}\) are examples of function spaces where each member is a differentiable function up to order \(k\) or \(k-1\), respectively, on the interval \([0, 1]\).


Key function spaces often encountered include:

• \(C[0, 1]\): The space of continuous functions on \([0, 1]\).
• \(C^k[0, 1]\): The space of functions that are \(k\)-times continuously differentiable on \([0, 1]\).
• \(L^p[0, 1]\): The space of functions whose \(p\)-th power is integrable on \([0, 1]\).

Function spaces equip these sets with norms that enable the measurement of the 'size' of functions, making it possible to study their convergence, continuity, and more.

In our problem, we specifically looked at the norms on \(C^k[0, 1]\), defined by summing the maximum absolute values of the derivative terms. This norm helps us in analyzing the behavior of functions and ensuring that certain properties, like boundedness and equicontinuity, can be checked to apply results like the Arzelà–Ascoli theorem. By carefully examining these properties, we successfully showed the compactness of the identity map from \(X_k\) to \(X_{k-1}\).