# Tagged Questions

For question involving Hilbert spaces, that is, complete normed spaces whose norm comes from an inner product.

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### Orthogonal complement of a single element of Hilbert space

Let $F = \{e\}$ be a set, where $\|e\| = 1$. My question is, what is $P_{F^\perp}h$ for all $h$ in a Hilbert space? Another question is, does $e$ all by itself, make up an ortonormal system?
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### Having trouble showing a subspace of $\ell^2$ is closed.

Let $M =\{ (x_n)_{n \in \mathbb{N}} \in \ell^{2} \mid x_{2j} = 0, \text{ for all } j \in \mathbb{N}\}$. $M$ is a subspace of the Hilbert space $\ell^{2}$ and I'm supposed to show it's closed. ...
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### Questions about Hilbert spaces, linear subspaces and orthonormal bases

I've been looking over some old assignments in my analysis course to get ready for my upcoming exam - I've just run into something that I have no idea how to solve, though, mainly because it looks ...
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### The domain of a root of a self-adjoint operator associated with an interpolation space

We now that $V$, $H$ are separable Hilbert spaces such that $V$ is dense in $H$ and $V\hookrightarrow H$ continuous, by representation theorem exists $A: D(A)\subset V\rightarrow H$ self adjoint e ...
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### Compatible Hilbert space subspaces - need help understanding a statement made in a book

A book I'm reading has the following in a section on lattices formed by subspaces of a Hilbert space : Two subspaces $M$ and $N$ are compatible if there exist three mutually disjoint subspaces ...
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### What does it mean for two functions to be orthogonal? [duplicate]

When two finite dimensional vectors are orthogonal, i.e. perpendicular, their dot product is exactly zero, e.g. $$\mathbf{a}\cdot\mathbf{b}=a_1b_1+\cdots+a_nb_n=0.\tag{1}$$ When I studied functional ...
### Every complete orthonormal set in a Hilbert space $H$ is an orthonormal basis, if and only if $H$ is finite dimensional.
Show that any orthonormal set in a Hilbert space $H$ is linearly independent, and use this to show that $H$ is finite dimensional if and only if every complete orthonormal set is an orthonormal basis. ...