Derivative of norm in Banach Spaces

Question

I was looking at this, and saw the author used the inequality

$$\frac{d}{dt}||x(t)||\leq ||x'(t)||$$ where $x:[a,b]\to X$ is a function of class $C^1$, and $X$ is a Banach space. I can easily prove the inequality if there is an inner product involved (Using Cauchy-Schwarz), but in general, as it is a Banach space, there may not be one.

How can I prove the inequality if $X$ is not a Hilbert space, I mean, the norm has not an associated inner product? Is it much harder? Is it really true? Or... how can I define the derivative of the norm if there is no inner product involved?

Answer 1

Assuming that the derivative on the left hand side exists, it is the limit of

$$\frac{||x(t+d)||-||x(t)||}{d}$$

The numerator is less than $||x(t+d)-x(t)||$ so the absolute value of the fraction is bounded by $||\frac{x(t+d)-x(t)}{d}||$.

which converges to the norm of the derivative of x.

Answer 2

Hint: use $$ \|x(t+h)\|-\|x(t)\|\le\|x(t+h)-x(t)\| $$ and the definition of derivative.