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# Schauder bases

Whereas the concept of a Hamel basis is very general — it applies to any vector space — it is not particularly well suited for infinite-dimensional Banach spaces.

### › Infinite-dimensional Banach spaces have only uncountable Hamel bases

Let $X$ be an infinite-dimensional Banach space. Then a sequence $\{e_j\}_{j\in \mathbb N}$ cannot be a Hamel basis for $X$.

## Schauder Bases

Let $(X, \|\cdot\|)$ be a Banach space. A sequence $\{e_j\}_{j\in \mathbb N}$ is called a Schauder basis (or countable basis) for $X$ if every vector $x \in X$ has a unique represention $x = \sum_{j\in \mathbb N} x_j e_j,$ meaning that $\lim_{N \to \infty}\| x- \sum_{j=1}^N x_j e_j\| = 0.$ The scalars $x_j$ are the coordinates of $x$.

N.b. Any space with a Schauder basis is separable.1)

From now on, the word basis refers to an ordered basis, Hamel (in the case of any finite-dimensional vector space) or Schauder (in the case of any infinite-dimensional Banach space). In both cases, a basis assigns to each $x \in X$ unique coordinates $x_1, x_2, \ldots$.

Ex.
• Let $e_j = (0,\ldots,\underbrace{1}_{\text{jth position}},0 \ldots)$. Then $\{e_j\}_{j=1}^\infty$ is a (Schauder) basis for $l_p$, $1 \leq p < \infty$:2)
Approximation property:$x = \{x_j\}_{j\in \mathbb N} \in l_p \quad\Longrightarrow\quad \sum_{j = 1}^\infty |x_j|^p < \infty \quad\Longrightarrow\quad \sum_{N+1}^\infty |x_j|^p \to 0 \text{ as } N \to \infty.$ Thus $\big\| \sum_{j=1}^N x_j e_j - x \big\|_{l_p} = \| (x_1, \ldots, x_N, 0, 0, \ldots) - (x_1,\ldots,x_N, x_{N+1}, \ldots)\|_{l_p} = \big( \sum_{N+1}^\infty |x_j|^p \big)^{1/p} \to 0 \text{ as } N \to \infty.$ Uniqueness of coordinates: $\sum_{j = 1}^\infty x_j e_j = \sum_{j=1}^\infty y_j e_j \quad\Longleftrightarrow\quad \lim_{N \to \infty}\sum_{j=1}^N |x_j - y_j|^p = 0 \quad\Longrightarrow\quad x_j = y_j, \text{ for all } j \in \mathbb N.$
• The trigonometric functions $\{\mathrm{e}^{ikx}\}_{k \in \mathbb Z}$ is a (Schauder) basis for $L_2((-\pi,\pi),\mathbb C)$. The coordinates in this basis are known as Fourier coefficients.
• The vectors $(1,0,0), (1,1,0),(1,1,1)$ provide a (Hamel) basis for $\R^3$. Find the coordinates $[c_1,c_2,c_3]$ of $(2,0,1)$ in this basis.$c_1 \begin{bmatrix} 1\\0\\0 \end{bmatrix} + c_2 \begin{bmatrix} 1\\1\\0 \end{bmatrix} + c_3 \begin{bmatrix} 1\\1\\1 \end{bmatrix} = \begin{bmatrix} 2\\0\\1 \end{bmatrix} \quad\Longleftrightarrow\quad \begin{bmatrix} c_1 + &c_2 + &c_3 \\ &c_2 + &c_3 \\ &&c_3 \end{bmatrix} = \begin{bmatrix} 2\\0\\1 \end{bmatrix} \quad\Longleftrightarrow\quad \begin{bmatrix} c_1 \\ c_2 \\ c_3\end{bmatrix} = \begin{bmatrix} 2\\-1\\1\end{bmatrix}.$ The coordinates for $(2,0,1)$ in the new basis are $[2,-1,1]$.
1)
The opposite is not true; there are (strange) separable Banach spaces with no Schauder basis.
2)
$l_\infty$ has no Schauder basis.