Question about definition of inner product space

Question

While practising I came across the following easy question:

"Is the space $B(0,1):=\{f:[0,1]\rightarrow\mathbb{R}$ bounded$\}$ an inner product space?"

But I'm not quite sure what the correct answer is here. As far as I can tell this is not even a space but just a set. No operations or inner products are defined on it, so it does not have any structure. To me this seems like asking

"Is the space $\{a,b,c\}$ an inner product space?"

Which without any further information seems like it's a strange question.

So my question is:

1. Why do they even call this a space while the specification is just a set?
2. What is the correct answer here?

Answer 1

You can easily see that the set $B(0,1)$ is closed under scalar multiplication, and $+ , \times $. So it's a Banach space (with the sup norm) and also an algebra.

Put inner product $\langle f,g\rangle = \int f(x)\bar g(x) dx$, $B(0,1)$ is an inner product but not But a Hilbert space, because it's not complete.