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I am reading ergodic theory notes. While reading about the mean ergodic theorem of von Neuman, I was perplexed by the following statement:

" In the $L^2$ case it was projection onto a certain subspace, but since $L^1$ is not a Hilbert space, we can’t make sense of “projection operators” as we did before"

Why are projection operators relevant only in a Hilbert space? And is there an example for why those kind of operators are not relevant in non-Hilbert spaces? ($L^{1}$ for example)?

(Here are the notes I follow https://www.mit.edu/~fengt/ergodic_theory.pdf)

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    $\begingroup$ Orthogonal projection is defined by orthogonality, which is defined using the inner product. $\endgroup$
    – Jürgen Sukumaran
    Commented Jul 13, 2022 at 9:37
  • $\begingroup$ But don't we have the same inner product defined in $L^{1}$ also? $\endgroup$
    – user2582354
    Commented Jul 13, 2022 at 9:49
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    $\begingroup$ It seems like the term "projection" is implicitly used as "orthogonal projection" in the linked notes, which I would say is technically wrong. Then it really makes sense to have "[orthogonal] projection" only in a space with inner product. Projections and projection operators makes sense in many contexts different from Hilbert spaces. $\endgroup$
    – Korf
    Commented Jul 13, 2022 at 9:56
  • $\begingroup$ Using a different naming convention isn't "technically wrong" @Korf especially when the convention is a common one. $\endgroup$
    – Jürgen Sukumaran
    Commented Jul 13, 2022 at 10:34
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    $\begingroup$ Perhaps it is helpful to note the following general fact: A Banach space is isomorphic to a Hilbert space iff every closed subspace is complemented. $\endgroup$
    – JWP_HTX
    Commented Jul 20, 2022 at 22:00

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It is probably because Hilbert's projection theorem is true only in Hilbert spaces! And he probably needs that. If $H$ is a subspace of $X$ (Hilbert) and $x$ not in $H$ then there is a unique element $m$ of $H$ such that $min_{H}$$\left\|y-x \right\|=\left\|m-x \right\|$ and $<m-x,y>=0$ for all $y$ in $H$. So projection, orthogonality and inner product are a family of properties only met in Hilbert spaces!!

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