Thinking of the intersection of the unit sphere in a Banach space $X$ with a subspace $L$ it made me think that this should give me the unit sphere in $L$. Yet someone told me I am thinking too simple here and that in Banach spaces this does not necessarily hold. I cannot find any references where this is treated, can anyone help me? Either an idea, an example or a reference would help.
My idea in Hilbert spaces was: Let $H$ be a Hilbert space and $S_H = \{x \in H~ |~ \|x\|_H = 1\}$. Then
$S_H\cap L = \{x\in L ~|~ \|x\|_H = 1\} \overset{?}{=}\{x\in L ~|~ \|x\|_L =1 \}= S_L$, where $\|\cdot\|_L$ is the induced norm from $H$, is the unit sphere in $L$.
So in a Hilbert space where the ball is "equally rotund" this should equal the same set. But how about in Banach spaces? Does this only give me a set such that there is a norm which makes this set the unit sphere? What property do I actually need here?
In case it helps with finding an answer: I am particularly interested in the intersection of spheres and balls from general Banach spaces with 2-dimensional planes.