Una decomposìzioue di uno spazio hilbertiano avente interesse e significato in Meccanica

Valent, Tullio

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 52 (1972) no. 2, pp. 182-186

Résumé

Let $\mu$ be a measure on $R^{n}$ and let $X$ be a bounded measurable subset of finite measure of $R^{n}$. We denote by $L^{2}_{R^{n}}(X,\mu)$ the Hilbert space of all $R^{n}$-valued functions square-integrable on $X$ with respect to $\mu$, where the scalar product is defined as the integral over $X$ of the ordinary product of two vectors. We show that $L^{2}_{R^{n}}(X,\mu)$ is the topological direct sum of two ortogonal subspaces, which have an evident mechanical meaning for $n=3$. We show then some consequences and applications of the preceding result.