I am supposed to show that $c_\infty \cong c_0 \oplus \mathbb{C}$, where $$ c_\infty := \{(a_n)_{n \in \mathbb{N}}| a_n \in \mathbb{C}, a_n \to a \in \mathbb{C}\}, \\ c_0 := \{(a_n)_{n \in \mathbb{N}}| a_n \in \mathbb{C}, a_n \to 0\}, $$ so $c_\infty$ is the space of convergent sequences with values in $\mathbb{C}$ and $c_0$ the space of null sequences. I am confused by a view things:
What does $c_0 \oplus \mathbb{C}$ mean? We defined a direct sum as follows: If $M$ and $N$ are linear subspaces of a linear space $X$, we write $X = M \oplus N$ iff $M+N = X$ and $M \cap N = \{0\}$. Since $\mathbb{C}$ is not a subspace from $\mathbb{C}^{\mathbb{N}}$, the direct sum does not make sense. Of course, one can identify $\mathbb{C}$ with the space of all constant sequences, say $c$. Then we have actually have $c_\infty = c_0 \oplus c$, right?
So, when we write $c_\infty \cong c_0 \oplus \mathbb{C}$, do we actually just mean $c \cong \mathbb{C}$ and $c_\infty = c_0 \oplus c$ or has $c_\infty \cong c_0 \oplus \mathbb{C}$ another precise mathematical meaning?