Suppose we have $\mathbb{R}$
Then imagine an interval in $\mathbb{R}$, let's call it $I$.
Then $I$ is not a subspace, since addition of two elements do not necessarily lie in the set.
It seems to me that the only subspace in $\mathbb{R}$ is the set $\{0\}$.
Likewise, if we take some arbitrary chunk from $\mathbb{R}^n$, it might not be a subspace. But others might, e.g., a hyperplane.
So, geometrically speaking, how do we generate set in $\mathbb{R}^n$ that are subspaces?
Do we always visualize it as some line, plane, passing through the origin, but not other objects such as a sphere, or a ball? Why is this true?
If $V$ is a subspace of $\mathbb R^n$ and $V\neq\{0\}$, then there is some $v\in V$ such that $v\neq0$. But then the line $\{\lambda v\mid\lambda\in\mathbb R\}$ is a subset of $V$. So, no ball is a subspace of $\mathbb R^n$ and the same argument applies to spheres. Actually, the same argument applies to any bounded subset $\mathbb R^n$ other than $\{0\}$.