# The concept/physical meaning/interpretations behind the Bessel's inequality

If $\{\boldsymbol{\varphi}_1, \boldsymbol{\varphi}_2, ... \}$ is an orthonormal system in $\mathcal{H}$, then the Bessel's inequality is:

$$\sum_{j=1}^{\infty} |\langle \mathbf{x},\boldsymbol{\varphi}_j \rangle|^2 \leq \lVert \mathbf{x} \rVert^2 \quad \text{for every } \mathbf{x}\in \mathcal{H}.$$

My question is:

• What is the meaning/concept of this inequality?
• Is there any geometric or physical interpretations about that?

• The idea is that, if you had a full basis, you would expect $x=\sum_j \langle x,\varphi_j\rangle\varphi_j$, which would then lead to $\|x\|^2=\sum_j |\langle x,\varphi_j\rangle|^2$. If the system is orthonormal, but not necessarily a full basis, then $\sum_j |\langle x,\varphi_j\rangle|^2 \le \|x\|^2$, because you might be missing components in the sum $\sum_j \langle x,\varphi_j\rangle \varphi_j$ needed to have a full expansion of $x$. – DisintegratingByParts Apr 17 '17 at 10:56
• Thanks @TrialAndError. Could you please explain about the orthonormal basis and full basis in $\mathcal{H}$? – Amin Apr 17 '17 at 11:18
• In my comment replace "full basis" in the first sentence with "full orthonormal basis". – DisintegratingByParts Apr 17 '17 at 11:29
• It can be a orthonormal basis of a subspace rather than the full space. – mathreadler Apr 17 '17 at 11:49

It's like the Pythagorean theorem, but we might not have enough vectors in our orthonormal system to represent $x$ exactly. If $x \in \mathbb R^3$ and we have an orthonormal set of vectors $\{\phi_1,\phi_2,\phi_3\} \subset \mathbb R^3$, then we can express $x$ as a linear combination $x = \sum_{j=1}^3 \langle x,\phi_j\rangle \phi_j$. The Pythagorean theorem then tells us that $\|x\|^2=\sum_{j=1}^3 | \langle x,\phi_j\rangle |^2$. If we had only two vectors in our orthonormal set, we might not be able to represent $x$ exactly, and we would get an inequality rather than an equality.
$$\text{if } {\bf u} \in V = \{{\bf v_1,v_2,\cdots,v_n}\}$$ $$\text{then there exist unique } c_k \text{ so that } {\bf u} = \sum_{k=1}^{n} c_k{\bf v}_k$$