# Bessel inequality

The inequality

$$\| f \| ^ {2} = (f, f) \geq \ \sum _ {\alpha \in A } \frac{| (f, \phi _ \alpha ) | ^ {2} }{( \phi _ \alpha , \phi _ \alpha ) } =$$

$$= \ \sum _ {\alpha \in A } \left | \left ( f, \frac{\phi _ \alpha }{\| \phi _ \alpha \| } \right ) \right | ^ {2} ,$$

where $f$ is an element of a (pre-) Hilbert space $H$ with scalar product $(f, \phi )$ and $\{ {\phi _ \alpha } : {\alpha \in A } \}$ is an orthogonal system of non-zero elements of $H$. The right-hand side of Bessel's inequality never contains more than a countable set of non-zero components, whatever the cardinality of the index set $A$. Bessel's inequality follows from the Bessel identity

$$\left \| f - \sum _ {i = 1 } ^ { n } x ^ {\alpha _ {i} } \phi _ {\alpha _ {i} } \ \right \| ^ {2} \equiv \ | f | ^ {2} - \sum _ {i = 1 } ^ { n } \lambda _ {\alpha _ {i} } | x ^ {\alpha _ {i} } | ^ {2} ,$$

which is valid for any finite system of elements $\{ {\phi _ {\alpha _ {i} } } : {i = 1 \dots n } \}$. In this formula the $x ^ {\alpha _ {i} }$ are the Fourier coefficients of the vector $f$ with respect to the orthogonal system $\{ \phi _ {\alpha _ {1} } \dots \phi _ {\alpha _ {n} } \}$, i.e.

$$x ^ {\alpha _ {i} } = \ { \frac{1} \lambda _ {\alpha _ {i} } } (f, \phi _ {\alpha _ {i} } ),\ \ \lambda _ {\alpha _ {i} } = \ ( \phi _ {\alpha _ {i} } , \phi _ {\alpha _ {i} } ).$$

The geometric meaning of Bessel's inequality is that the orthogonal projection of an element $f$ on the linear span of the elements $\phi _ \alpha$, $\alpha \in A$, has a norm which does not exceed the norm of $f$( i.e. the hypothenuse in a right-angled triangle is not shorter than one of the other sides). For a vector $f$ to belong to the closed linear span of the vectors $\phi _ \alpha$, $\alpha \in A$, it is necessary and sufficient that Bessel's inequality becomes an equality. If this occurs for any $f \in H$, one says that the Parseval equality holds for the system $\{ {\phi _ \alpha } : {\alpha \in A } \}$ in $H$.

For a system $\{ {\phi _ \alpha } : {\alpha = 1, 2 , . . . } \}$ of linearly independent (not necessarily orthogonal) elements of $H$ Bessel's identity and Bessel's inequality assume the form

$$\left \| f - \sum _ {\alpha , \beta = 1 } ^ { n } b _ {n} ^ {\alpha \beta } (f, \phi _ \beta ) \phi _ \alpha \right \| ^ {2\ } \equiv$$

$$\equiv \ \| f \| ^ {2} - \sum _ {\alpha , \beta = 1 } ^ { n } b _ {n} ^ {\alpha \beta } (f, \phi _ \alpha ) (f, \phi _ \beta ),$$

$$\| f \| ^ {2} \geq \sum _ {\alpha , \beta = 1 } ^ { n } b _ {n} ^ {\alpha \beta } (f, \phi _ \alpha ) (f, \phi _ \beta ),$$

where $b _ {n} ^ {\alpha \beta }$ are the elements of the matrix inverse to the Gram matrix (cf. Gram determinant) of the first $n$ vectors of the initial system.

The inequality was derived by F.W. Bessel in 1828 for the trigonometric system.

#### References

 [1] L.D. Kudryavtsev, "Mathematical analysis" , 2 , Moscow (1973) (In Russian)

Usually, the orthogonal system of elements $\{ \phi _ \alpha \}$ is orthonormalized, i.e. one sets $\psi _ \alpha = \phi _ \alpha / \| \phi _ \alpha \|$. Bessel's inequality then takes the form
$$\sum _ {\alpha \in A } | (f, \psi _ \alpha ) | \leq \ \| f \| ^ {2} ,$$