Linear subspace

Let $(X,+,\cdot)$ be a vector space over $K$, and let $Y\subset X$.

The set $Y$ is a linear subspace of $X$ if:

1. $0\in Y$,
2. $a+b\in Y$ for all $a,b\in Y$,
3. $\lambda a\in Y$ for all $\lambda\in K$ and $a\in Y$.

With the inherited operations, $Y$ is itself a vector space , and many constructions in analysis arise as subspaces (kernels, images, function spaces, etc.). Subspaces are also the building blocks for quotient spaces and direct sums .

Examples:

• $\{0\}$ and $X$ are subspaces of $X$.
• In the space of all sequences $s$, the set $\ell^1=\{x=(x_n):\sum_{n=1}^\infty |x_n|<\infty\}$ is a subspace.
• In $F([a,b])$, the set $C[a,b]$ of continuous functions is a subspace.