Definition of Graded Algebra

Question

I attempt to understand the definition of graded algebra from An Introduction to Manifolds by Loring Tu (page no. 30). Below I quote the definition supplied in the book.

An algebra $A$ over a field $K$ is said to be graded if it can be written as a direct sum $A = \bigoplus_{k = 0}^{\infty} A^k$ of vector spaces over $K$ such that the multiplication map sends $A^k \times A^l$ to $A^{k+l}$. The notation $A = \bigoplus_{k = 0}^{\infty} A^k$ means that each nonzero eleement of $A$ is uniquely a finite sum $$ a = a_{i_1} + \cdots a_{i_m}, $$ where $a_{i_j} \neq 0 \in A^{i_j}$.

My Questions

1. Why does the sum for $a = a_{i_1} + \cdots a_{i_n}$ have to be finite?
2. Why is the condition: $a_{i_j} \neq 0 \in A^{i_j}$ applied for the sum?

3. Can you please provide a simple example of the multiplication map that sends $A^k \times A^l$ to $A^{k+l}$ for a graded vector space $A$ over $K$?

4. The definition in the quote mentions a multiplication map that sends $A^k \times A^l$ to $A^{k+l}$. According to my understanding, here a multiplication map $\mu: A \times A \to A$ can be considered which preserves the degree; i.e., $\mu(A^k, A^l) \subseteq A^{k+l}$, where $k, l \in \mathbb{N}$. Does this multiplication map also satisfy distributivity, associativity, and homogeneity (just like the definition of algebra between two vector spaces over a field)?

For question no. 1 and 2, my confusion arises from the fact that I am trying to see these two conditions (finite sum and $a_{i_j} \neq 0 \in A^{i_j}$) as consequences of the direct sum of vector spaces, not just as parts of the definition, in particular as concrescences of the following definition and theorem.

Definiton (Direct sum of vector spaces): Let $U$ and $V$ be subspaces of a vector space $W$. If the sum $U + V = W$ and $ U \cap V = \bar{0}$, then $W$ is said to be the direct sum of $U$ and $V$, where $\bar{0}$ is the zero vector in $U$, $V$ and $W$. In this case, we write $W = U \bigoplus V$.

Theorem: Every element $w \in W$ can be written uniquely as $w = u + v$, where $u \in U$ and $v \in V$.

In this definition and theorem, the conditions described in question no. 1 and 2 are not mentioned.