A manifold on the real commutative Banach algebra $C([0,1];\mathbb R)$ that cannot be embedded in the finite-dimensional Euclidean space $C([0,1];\mathbb R^n)

 $C([0,1];\mathbb R)$ and $\mathbb R^2 \, ( \, = \, C(\{0,1\};\mathbb R) \, )$ are simple examples of a real commutative Banach algebra. $C([0,1];\mathbb C)$ and $\mathbb C^2 \, ( \, = \, C(\{0,1\};\mathbb C) \, )$ are simple examples of a commutative $C^*$-algebra. Here, we consider $C([0,1];\mathbb R)$, which is the set of all real-valued continuous functions on the bounded closed interval $[0,1]$. The interval is a contractible compact Hausdorff space. The direct product space $(C([0,1];\mathbb R))^n \, ( \, = \, C([0,1];\mathbb R^n) \, )$ is a real Banach space and a free $C([0,1];\mathbb R)$-module. A $C^1$-mapping from an open set of $C([0,1];\mathbb R^n)$ to $C([0,1];\mathbb R)$
is said to be $C([0,1];\mathbb R)$-smooth, if its Frechet derivatives are $C([0,1];\mathbb R)$-linear. Then, we can define a concept of an $n$-dimensional smooth $C([0,1];\mathbb R)$-manifold.

 In this memo, we see existence of a connected metrizable $1$-dimensional
smooth $C([0,1];\mathbb R)$-manifold that cannot be embedded in $C([0,1];\mathbb R^n)$. In Section 1, as a primitive observation, we see that embeddability
in the Cartesian space $C([0,1];\mathbb R^n)$ as a smooth $C([0,1];\mathbb R)$-submanifold implies existence of an analog of a bump function. Then, in Section 2, we construct an example where no such analog exists.