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In the introduction of the book lambda calculus with types by Barendregt, there are examples of typed functions like enter image description here

And I wonder what operation the underlined type expression is for.

Although we can surely make an operation of the type, I want to find the real example in functional analysis that the author was thinking about.

The cited book is Lax, Functional Analysis.

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prime235711
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  • I'm not particularly interested in trying to untangle this at the moment, but maybe some of the possibilities in this answer will work, or can be adjusted to work, or at least indicate other possibilities. – Dave L. Renfro Aug 06 '23 at 13:18
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    The underlined type seems to be of the form $() \rightarrow () \rightarrow ()$. Do we have to know a certain convention about how this is associated? – GEdgar Aug 06 '23 at 13:34
  • Yes. In this area, it is usually right-associative. So the function takes two arguements. (Or if the rightmost parenthesis is discarded, it takes three arguements.) – prime235711 Aug 07 '23 at 11:54

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