In a book I'm reading (Functional Analysis for Probability and Stochastic Processes by Adam Bobrowski) Exercise 2.2.13 is as follows:
Suppose that $\mathbb{Y}$ is a subspace of a Banach space $\mathbb{X}$. Show that $\mathbb{Y}$ is itself a Banach space, equipped with the norm inherited from $\mathbb{X}$.
Obviously every closed subspace of a Banach space is also a Banach space but if we do not required the subspace to be closed (which is the case in the exercise), then I think this claim is wrong.
Take for example the space $(L^{1}([0,1]),\|\cdot\|_{1})$ with $\|f\|_{1}:=\int_{[0,1]}|f|d\lambda(x)$ which is known to be a Banach space. The space $(C([0,1]),\|\cdot\|_{1})$ is a subspace of $L^{1}([0,1])$. However $C([0,1])$ is the canonical example of a vector space which is not a Banach space.
Is there something I'm missing or is the exercise simply wrong?