Question

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

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

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.

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.

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 \([0,1]\).

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).

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.

Key concepts

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 \([0,1]\).