1. ## Norm on C^1[a,b]

Let $C^{1} [a,b]$ be the space of differentiable real valued functions $f:[a,b] \to \mathbb {R}$ such that the derivative $f':[a,b] \to \mathbb {R}$ is continuous. Given $f \in C^{1} [a,b]$ define $||f||_{C^{1}}$ by

$||f||_{C^{1}}=||f||_{\infty} + ||f'||_{\infty}$.

(a) Prove that $||-||_{C^{1}}$ is a norm on $C^{1} [a,b]$.

Ok, so I have absolutely no idea how to do this. What defines a norm on a space of functions? How does one show that something is a norm?

(b) Prove that $C^{1} [a,b]$ is complete in this norm.

Now I know that completeness of a space is equivalent to every Cauchy sequence in the space being convergent.

So let $f_{n}$ in $C^{1}$ be a sequence of functions and suppose that $f_{n}$ is Cauchy.

So for every $\varepsilon > 0$ there is an N such that $n, l \geq N \implies ||f_{n} - f_{l}|| < \varepsilon$.

Need to show that $f_{n}$ is convergent.

Now this is where I get confused. How does the norm relate to any of this? What does it mean for a space to be complete in a norm?

2. Originally Posted by nmatthies1
Let $C^{1} [a,b]$ be the space of differentiable real valued functions $f:[a,b] \to \mathbb {R}$ such that the derivative $f':[a,b] \to \mathbb {R}$ is continuous. Given $f \in C^{1} [a,b]$ define $||f||_{C^{1}}$ by

$||f||_{C^{1}}=||f||_{\infty} + ||f'||_{\infty}$.

(a) Prove that $||-||_{C^{1}}$ is a norm on $C^{1} [a,b]$.

Ok, so I have absolutely no idea how to do this. What defines a norm on a space of functions? How does one show that something is a norm?
Do you know the definition of a norm? Norms are defined on vector spaces and C^1 is a vector space (by pointwise addition) over the reals.
(b) Prove that $C^{1} [a,b]$ is complete in this norm.

Now I know that completeness of a space is equivalent to every Cauchy sequence in the space being convergent.

So let $f_{n}$ in $C^{1}$ be a sequence of functions and suppose that $f_{n}$ is Cauchy.

So for every $\varepsilon > 0$ there is an N such that $n, l \geq N \implies ||f_{n} - f_{l}|| < \varepsilon$.

Need to show that $f_{n}$ is convergent.

Now this is where I get confused. How does the norm relate to any of this? What does it mean for a space to be complete in a norm?
A norm is what determines if a space is complete. Completeness is a property of a metric, every norm induces a metric given by $d(x,y)=||x-y||$. Now if you change a metric (or a norm), the space may not be complete. Take the metric $d(x,y)=|\frac{1}{x}-\frac{1}{y}|$ on $[1,\infty)$ (which is complete under the Euclidian metric). Under this metric the sequence of natural numbers is Cauchy, but it coverges to 0 which is not in the set.

As for the solution, what can you say about $||f_n-f_m||_\infty$ and $||f'_n-f'_m||_\infty$. What happens to Cauchy sequences in $(C^1[0,1],||\cdot||_\infty)$?