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What would be the most intuitive way to perceive a closed subspace of Hilbert space. We know that any subspace of a finite-dimensional Hilbert space is closed.

Could we use the simplest analogy with $\Bbb{R}^2$, which Hilbert space, and say that every line which passes through the center of our space is closed subspace?

user729424
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Maksim Tomic
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  • I don't think there's an intuitive way to perceive non-closed subspaces of Hilbert (or normed) spaces. That's because the phenomenon occures in infinite dimension, which is far from our intuition. The line analogy is too simple. Lines (as in 1 dimensional subspaces) are closed in any normed vector space, finite dimensional or not. In fact for a linear subspace to be non-closed both the space and the subspace have to have infinite dimension. Obviously this is necessary but not a sufficient condition. – freakish Apr 10 '20 at 08:15
  • What Hilbert spaces are you familiar with? Continuous functions or polynomials are non-closed subspaces of $L^2([0,1])$. On the other hand, functions with zero average are a closed subspace. Closed subspaces are pretty much analogous to lines and planes through the origin in 3D space, non-closed ones are not, they are dense in some such closed subspace. If we drop the requirement that spaces are over reals than rationals are such a dense non-closed subspace of reals. – Conifold Apr 10 '20 at 08:26
  • In a Hilbert space, a subset $S$ is a closed subspace if and only if $S={s\mid\forall x\in X:\langle s,x\rangle=0}$ for some subset $X$. – Berci Apr 10 '20 at 09:15

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