Let $(C, \|\cdot\|)$ be the Banach space of continuous paths $x: [0,1]\rightarrow\mathbb{R}^d$ starting at zero with sup-norm $\|\cdot\|$.

Let further $B\subset C$ be the subspace of $0$-started continuous paths of bounded variation, i.e.

$$B = \big\{x\in C \ \big| \ \|x\|_1:=\sup\nolimits_{(t_\nu)}{\textstyle\sum_\nu}|x_{t_\nu} - x_{t_{\nu-1}}|<\infty \ \text{ and } \ x(0)=0\big\}$$

where the above sup runs over all partitions $(t_\nu)$ of $[0,1]$.

The set $B$ can be endowed with both the (uniform) norm $\|\cdot\|$ and the (1-variation) norm $\|\cdot\|_1$.

Do you know if the Borel $\sigma$-algebras on $(B, \|\cdot\|)$ and $(B, \|\cdot\|_1)$ coincide?

Any references or hints are welcome.

*Edit:* As pointed out by Gerald Edgar, we narrowed $B$ down to paths started at zero to make the question more meaningful.

*Edit 2:* As Gerald Edgar pointed out in the comments to his answer below, the answer to this question is **no, the Borel $\sigma$-algebras cannot coincide** because there are more $\|\cdot\|_1$-open sets in $B$ than there are $\|\cdot\|$-Borel sets in $C$.