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Chapter 1: Problem 14

Let \(C^{n}[0,1]\) be the space of all real-valued functions on \([0,1]\) that have \(n\) continuous derivatives on \([0,1]\), with the norm $$ \|f\|=\max _{0 \leq k \leq n}\left(\max \left\\{\left|f^{k}(t)\right| ; t \in[0,1]\right\\}\right) $$

Short Answer

Expert verified
Calculate all derivatives of \(f\) up to \(f^{(n)}\), find their maximum absolute values over \([0,1]\), and take the largest of these values.

Step by step solution

01

Understanding the Problem

Identify the space and the norm in question. The space given is denoted as \(C^{n}[0,1]\), which consists of all real-valued functions on \([0,1]\) that possess \(n\) continuous derivatives on the interval \([0,1]\). The norm involves taking the maximum absolute value of each of these derivatives over the interval.
02

Interpret the Norm

The norm \(orm{f}\) is given by \(orm{f}=\text{max}_{0 \leq k \leq n} \text{max} \{orm{f^{(k)}(t)}; t \in [0,1]\}\). This means to find the norm of a function \(f\) in this space, find the absolute maximum value of each of \(f\)'s derivatives from 0 up to \(n\) across the interval \([0,1]\), and then take the highest of these maximum values.
03

Determine the Derivatives

Calculate the derivatives of the function \(f\) from 0 up to \(n\). This involves first finding \(f^{(0)}(t) = f(t)\), then \(f^{(1)}(t) = f'(t)\), all the way up to \(f^{(n)}(t)\). Ensure all these derivatives exist and are continuous on \([0,1]\).
04

Find Maximum Values

For each derivative from \(f^{(0)}\) to \(f^{(n)}\), find the maximum absolute value over the interval \([0,1]\). This can be done by evaluating the function and its derivatives at critical points within the interval, which includes the endpoints 0 and 1 and any points where the derivative is zero or does not exist (within the domain).
05

Compute the Norm

Take the largest value from the list of maximum absolute values of \(f\) and its derivatives. Thus, the norm \(orm{f}\) is the maximum of these values.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Normed Spaces
Normed spaces are fundamental in functional analysis. A normed space is a vector space equipped with a function called a norm. The norm measures the 'size' or 'length' of vectors in the space.
For a given vector space \(V\) over a field \(\mathbb{F}\) (real or complex numbers), a norm \( \| \cdot \| \) assigns a non-negative real number to each vector in \(V\). This number represents the magnitude of the vector. Norms must satisfy the following properties:
  • **Positivity**: \(\|v\| \geq 0 \) for all \(v \in V\) and \(\|v\| = 0 \) if and only if \(v = 0\).
  • **Scalability**: \(\|\alpha v\| = |\alpha| \|v\| \) for any scalar \(\alpha\) and any vector \(v \in V \).
  • **Triangle Inequality**: \( \|u + v\| \leq \|u\| + \|v\| \) for all \(u, v \in V \).
The concept of normed spaces helps in generalizing the idea of distances and magnitudes from Euclidean spaces to more abstract vector spaces.
In the context of the exercise, the space \(C^{n}[0,1]\) is a normed space where the norm is defined as the maximum value of the absolute values of the function and its derivatives up to order \(n\). This specialized norm provides a way to measure the 'size' of functions in terms of their behavior over the interval \([0,1]\).
Continuous Derivatives
A function having continuous derivatives is crucial for ensuring smooth and predictable behavior. If a function \(f\) is said to have continuous derivatives up to order \(n\), it means:
  • The function \(f\) itself is continuous.
  • Its first derivative \( f^{(1)}(t) = f'(t) \) is continuous.
  • Its second derivative \( f^{(2)}(t) = f''(t) \) is continuous.
  • And so on, up to its \(n^{th}\) derivative \( f^{(n)}(t) \) being continuous.
This smoothness is important in functional spaces like \( C^{n}[0,1] \), as it guarantees that the function and its derivatives do not have any abrupt changes over the interval \([0,1]\).
In the exercise, the requirement for \( n \) continuous derivatives ensures that we can meaningfully talk about the maximum values of the function and its derivatives without worrying about discontinuities making these maxima unrepresentative of overall behavior.
Maximum Norm
The maximum norm, also known as the sup norm or infinity norm, measures the greatest absolute value of a function over a given domain. For a function space, it extends this idea to consider the function and its derivatives. In the exercise, the norm \(\|f\| \) is given by:
\[ \|f\| = \max_{0 \leq k \leq n} \left( \max \left\{ \|f^{(k)}(t)\| ; t \in [0,1] \right\} \right) \] This means:
  • Find the absolute maximum value of the function \(f(t)\) over the interval \([0,1]\).
  • Then find the maximum values for all its derivatives: \(f'(t), f''(t), ..., f^{(n)}(t)\).
  • Take the highest of all these maximum values as the norm.
The maximum norm is useful in various applications because it focuses on the extreme behavior of the function and its derivatives, providing a strict and absolute measure of 'size.'
For example, with \(C^{n}[0,1]\), you're not just looking at how large the function gets but also how large its rates of change can get up to the nth derivative. This approach offers a comprehensive view of the function's behavior in the interval \([0,1]\).

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Most popular questions from this chapter

Let \(C\) be a compact set in a finite-dimensional Banach space \(X\). Show that \(\operatorname{conv}(C)\) is compact. Hint: If a point \(x\) lies in the \(\operatorname{conv}(E) \subset \mathbf{R}^{n}\), then \(x\) lies in the convex hull of some subset of \(E\) that has at most \(n+1\) points. Indeed, assume that \(r>n\) and \(x=\sum t_{i} x_{i}\) is a convex combination of some \(r+1\) vectors \(x_{i} \in E\). We will show that then \(x\) is actually a convex combination of some \(r\) of these vectors. Assume that \(t_{i}>0\) for \(1 \leq i \leq r+1\). The \(r\) vectors \(x_{i}-x_{r+1}\) for \(1 \leq i \leq r\) are linearly dependent since \(r>n\). Thus, there are real numbers \(a_{i}\) not all zero such that \(\sum_{i=1}^{r+1} a_{i} x_{i}=0\) and \(\sum_{i=1}^{r+1} a_{i}=0 .\) Choose \(m\) so that \(\left|a_{i} / t_{i}\right| \leq\left|a_{m} / t_{m}\right|\) for \(1 \leq i \leq r+1\) and define \(c_{i}=t_{i}-\frac{a_{i} t_{m}}{a_{m}}\) for \(1 \leq i \leq r+1\). Then \(c_{i} \geq 0, \sum c_{i}=\sum t_{i}=1, x=\sum c_{i} x_{i}\), and \(c_{m}=0\) Having this, we can use the fact that in \(\mathbf{R}^{n}, \operatorname{conv}(C)\) is an image of the compact set \(S \times C^{n+1}\) in \(\mathbf{R}^{n+1} \times C^{n+1}\), where \(S\) is formed by points \(\left\\{\lambda_{i}\right\\}_{1}^{n+1}\) such that \(\lambda_{i} \geq 0\) and \(\sum \lambda_{i}=1\), under the \(\operatorname{map}\left(\\{\lambda\\},\left\\{x_{i}\right\\}\right) \mapsto \sum \lambda_{i} x_{i}\).

Let \(A, B\) be convex compact sets in a Banach space \(X\). Show that \(\operatorname{conv}(A \cup B)\) and \(A+B\) are compact. Generalize this statement to a finite number of sets. Hint: Using Exercise \(1.5\), show that \(\operatorname{conv}(A \cup B)\) is a continuous image of the compact set \(\\{(\alpha, \beta) ; \alpha, \beta \geq 0, \alpha+\beta=1\\} \times A \times B\), so it is compact. Similarly, \(A+B\) is the image of \(A \times B\) under the continuous map \((x, y) \mapsto x+y\).

Let \(A\) be a subset of a Banach space \(X .\) Denote by sconv \((A)\) the set of all \(x \in X\) that can be written as \(x=\sum_{i=1}^{\infty} \lambda_{i} x_{i}\), where \(x_{i} \in A, \lambda_{i} \geq 0\), and \(\sum \lambda_{i}=1\). Show that if \(A\) is bounded, then sconv \((A) \subset \overline{\operatorname{conv}}(A)\) Let \(A\) be the set of all standard unit vectors \(e_{i}\) in \(\ell_{2} .\) Show that \(0 \in\) \(\overline{\operatorname{conv}}(A)\) and \(0 \notin \operatorname{sconv}(A)\) Hint: If \(\lambda_{i} \geq 0\) and \(\sum_{i=1}^{\infty} \lambda_{i}=1\), approximate by a finite linear combination given by \(\tilde{\lambda}_{1}=\lambda_{1}, \ldots, \tilde{\lambda}_{n}=\lambda_{n}, \tilde{\lambda}_{n+1}=1-\sum_{i=1}^{n} \lambda_{i}\) If \(\sum \lambda_{i} e_{i}=0\), then \(\lambda_{i}=0\) for every \(i\)

Let \(f \in L_{p_{0}}[0,1]\) for some \(p_{0}>1 .\) Show that \(\lim _{p \rightarrow 1^{+}}\|f\|_{L_{p}}=\|f\|_{L_{1}} .\) If \(f \in L_{\infty}[0,1]\), then \(\lim _{p \rightarrow \infty}\|f\|_{L_{p}}=\|f\|_{L_{\infty}}\) Let \(x \in \ell_{q}\) for some \(q \geq 1\). Show that \(\lim _{p \rightarrow \infty}\|x\|_{\ell_{p}}=\|x\|_{\ell_{\infty}}\).

Let \(H\) be a Hilbert space. Prove the generalized parallelogram equality: If \(x_{1}, \ldots, x_{n} \in H\), then $$ \sum_{\epsilon_{i}=\pm 1}\left\|\sum_{i=1}^{n} \varepsilon_{i} x_{i}\right\|^{2}=2^{n} \sum_{i=1}^{n}\left\|x_{i}\right\|^{2} $$ Hint: Induction on \(n\).
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