all elements of hilbert spaces with orthonormal bases can be written as sums of the basis elements

Theorem

If ${e_{n}}$ is an orthonormal basis of a hilbert space $H$ , then for all $u \in H$ we have

lim_{m \to \infty} \sum_{n = 1}^{m} ⟨ u, e_{n} ⟩ e_{n} = \sum_{n = 1}^{\infty} ⟨ u, e_{n} ⟩ e_{n} = u

Note

Thus if we have an orthonormal basis, every element can be expanded in this series in terms of the basis elements (called a Bessel-Fourier series). And thus every separable Hilbert space has an orthonormal basis.

Note

ie like in finite-dimensional linear algebra, we can write any element as a linear combination of the basis elements. But in this space, there might be an infinite number of such elements.

Proof (via Bessel's inequality)

First, we prove ${\sum_{n = 1}^{m} ⟨ u, e_{n} ⟩ e_{n}}_{m}$ is Cauchy. Let $ϵ > 0$ . Then by Bessel's inequality, we have

\sum_{n=1}^\infty \lvert \langle u, e_{n} \rangle \rvert {#2} \leq \lvert \lvert u \rvert \rvert {#2} < \infty

Thus, for all $M \in N$ such that for all $N \geq M$ we have $\sum_{m = N + 1}^{\infty} | ⟨ u, e_{n} ⟩ |^{2} < ϵ$

Then for all $m > ℓ \geq M$ we compute

\begin{align} \left\lvert \left\lvert \sum_{n=1}^m \langle u, e_{n} \rangle e_{n} - \sum_{n=1}^\ell \langle u, e_{n} \rangle e_{n} \right\rvert \right\rvert &= \sum_{n=\ell+1}^m \lvert \langle u, e_{n} \rangle \rvert {#2} \\ &\leq \sum_{\ell+1}^\infty \lvert \langle u, e_{n} \rangle \rvert {#2} \\ &< \epsilon^2 \end{align}

thus the sequence is indeed Cauchy. Since $H$ is complete, there exists some $\bar{u} \in H$ where

\bar{u} = lim_{m \to \infty} \sum_{n = 1}^{m} ⟨ u, e_{n} ⟩ e_{n}

By continuity of inner product, we know that for all $ℓ \in N$

\begin{aligned} ⟨ u - \bar{u}, e_{ℓ} ⟩ & = lim_{m \to \infty} ⟨ u - \sum_{n = 1}^{m} ⟨ u, e_{n} ⟩ e_{n}, e_{ℓ} ⟩ \\ = lim_{m \to \infty} [⟨ u, e_{ℓ} ⟩ - \sum_{n = 1}^{m} ⟨ u, e_{n} ⟩ ⟨ e_{n}, e_{ℓ} ⟩] \\ = ⟨ u, e_{ℓ} ⟩ - ⟨ u, e_{ℓ} \cdot 1 ⟩ \\ = 0 \end{aligned}

Thus $⟨ u - \bar{u}, e_{ℓ} ⟩ = 0$ for all $ℓ$ if and only if $u - \bar{u} = 0$ .

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all elements of hilbert spaces with orthonormal bases can be written as sums of the basis elements

References

See Also

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