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?

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.

  • $\begingroup$ I know all this, but this doesn't answer my question. $\endgroup$ – user2520938 Jan 17 '15 at 18:46
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    $\begingroup$ The vector space structure on $B(0,1)$ is implied by the use of the word "space". It's like saying "the group $\mathbb{Z}$". I didn't tell you what the group operation is, but of course you know what I mean; it's too cumbersome to say things like "the group $(\mathbb{Z}, +)$" all the time. The meaning of "Is $X$ an inner product space" is exactly "Does $X$ admit an inner product". $\endgroup$ – mollyerin Jan 17 '15 at 19:33
  • $\begingroup$ @mollyerin Oke thanks, that's more along the line of what I was looking for $\endgroup$ – user2520938 Jan 17 '15 at 21:48
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    $\begingroup$ I should mention that there is at least some subtlety here. For instance, the space $B(0,1)$ is often given the sup norm $||f||_{\infty} = \max_{x\in[0,1]} |f(x)|$, and in this context, the question "is $B(0,1)$ an inner product space" might mean "does $B(0,1)$ admit an inner product giving rise to the norm $||\cdot ||_{\infty}$". The problem is that there are many natural norms on $B(0,1)$, so it usually is good to specify which one is meant, and you're right to complain that this question doesn't. $\endgroup$ – mollyerin Jan 17 '15 at 22:38
  • $\begingroup$ @mollyerin Thanks again, that was even more along the lines of what I was looking for. $\endgroup$ – user2520938 Jan 18 '15 at 12:54

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