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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 \(x_1, x_2, \ldots\).

  • 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 \([2,-1,1]\).
The opposite is not true; there are (strange) separable Banach spaces with no Schauder basis.
\(l_\infty\) has no Schauder basis.
2017-03-24, Hallvard Norheim Bø