Space $\mathbb{R}^{[0,1]}$ with $x(0)=0$ is a separable Banach space

In Kuo's book Gaussian measures in Banach spaces, he says the space $$\mathbb{R}^{[0,1]}$$ the set of all real valued functions on $$[0,1]$$ with $$x(0)=0$$ is a separable Banach space under the norm $$\|x\|=\sup_{t}|x(t)|$$

I know that $$\mathbb{R}^{[0,1]}$$ is separable by The Engelking-Karlowicz theorem. But how can I show the space given above is open in it? Is this space really Banach?

The problem is, it is possible to define the Wiener process on $$\mathbb{R}^{[0,1]}$$ with a cylinder sigma algebra and show there exists a version on $$C[0,1]$$. I am just confused about the method used in the book.

• I don't see why $\sup_{t\in[0,1]} \lvert x(t)\rvert$ should be finite for a real-valued function $x$ (let even $x(0)=0$, for all it matters). Typically, $C[0,1]$ denotes the set of continuous real-valued functions, so perhaps there is a typo in the book. – Gae. S. Sep 11 '19 at 23:40
• Exactly why I'm asking. He doesn't say continuous – badatmath Sep 11 '19 at 23:40

The topological space $$\Bbb R^{[0,1]}$$ (which is also homeomorphic to the subspace of the functions such that $$x(0)=0$$) is not first-countable, therefore it is not metrisable. There must be a typo in the book, and $$C[0,1]$$ should stand for a space of continuous functions.