| Abstract: |
| We propose to survey the theory of nearness between operators acting on normed spaces and developed by S. Campanato at the end of the eighties in a series of papers. The aim of S. Campanato was to study existence and regularity results for some differential elliptic equations.
Obviously nearness is a reflexive relation.
The talk adresses the natural question of the \textbf{symmetry} of the nearness relation. We observe that when $(Y$ is an \textbf{inner product space} and $a$ is near $b$ for the constants $\alpha$ and $\kappa$, then $b$ is near $a$, but for the different
constants $\frac{1-\kappa^2}{\alpha}$ and $\kappa$.
When the dimension of $Y$ is greater or equal to three, then the
three following properties are equivalent: $Y$ is an inner product space, the Birkhoff-James orthogonality is symmetric, and the Campanato nearness is symmetric. |
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