Direct sum of Banach Spaces

Question

I want to show that direct sum of $c_0$ and $\mathbb K$ is isomorphic to $c_0$. Here $c_0$ is the Banach space of all sequences converge to $0$. Please give me a hint how to define the Isomorphism.

Answer 1

You can define $f:\mathbb{K}\oplus \mathbb{C_0}\longrightarrow \mathbb{C_0} $ such that $f\left(x_0,(x_1,x_2,x_3,\ldots ) \right)=(x_0,x_1,x_2,\ldots ) $. So f is linear and $\lVert f(x)\rVert =\sup \{ |x_0|,|x_1|,\ldots \} =\lVert x\rVert $