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

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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.

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Best Answer

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.

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