# What is the meaning of "Projection is relevant only in Hilbert Space"? Why it is non relevant in non-Hilbert space?

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)

• Orthogonal projection is defined by orthogonality, which is defined using the inner product. Commented Jul 13, 2022 at 9:37
• But don't we have the same inner product defined in $L^{1}$ also? Commented Jul 13, 2022 at 9:49
• 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.
– Korf
Commented Jul 13, 2022 at 9:56
• Using a different naming convention isn't "technically wrong" @Korf especially when the convention is a common one. Commented Jul 13, 2022 at 10:34
• 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. Commented Jul 20, 2022 at 22:00

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 $$=0$$ for all $$y$$ in $$H$$. So projection, orthogonality and inner product are a family of properties only met in Hilbert spaces!!