Definition of a countable direct sum of subspaces of a Banach space

Question

Let $X$ be a separable Banach space and $K\subseteq X$ a subspace. Let $\{H_i\}_{i\in I}$ be a countable collection of subspaces of $X$. Is it correct that $K=\bigoplus H_i$ iff every element $k\in K$ can be written uniquely as $k=\sum h_i$, where $h_i\in H_i$ for every $i\in I$? What is the exact definition of $\bigoplus H_i$ as a subspace of $X$?