# Tagged Questions

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

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### Every orthonormal set in a Hilbert space is contained in some complete orthonormal set.

Let $H$ be a Hilbert space. Show that every orthonormal set in $H$ is contained in some complete orthonormal set. I'm unable to start from any direction. Do I use the Gram-Schmidt orthogonalization ...
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### Countable basis of differentiable function

If $$B = \left \{ \mathbf{v}_{k} \right \}_{k=0}^{\infty}$$ is a countable basis of a differentiable function $\mathbf{u}$ where $$\mathbf{u} =\sum_{k=0}^{\infty}a_{i}\mathbf{v}_{k},$$ then show ...
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### Trace Class: Decomposition

This is only Q&A. Preview Trace class operators decompose. So proofs reduce to Hilbert-Schmidt! Problem Given a Hilbert Space $\mathcal{H}$. For the trace class: ...
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### Are Hilbert-Schmidt operators in non-separable Hilbert spaces compact?

The definition of Hilbert-Schmidt operator should still be valid even when the Hilbert space is not separable: If $e_i$ for $i\in I$ is an orthonormal basis for a Hilbert space, and ...
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### Restrictions of function decomposition in $R^3$

I'm interested in the properties of countable basis functions that span functions living in $\Bbb R^3$. Can I represent a $L^2$ normalizable function that has a point divergence, (for example, ...
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### Question about norm of linear operator's adjoint on a Hilbert space

Stein and Shakarchi state that for a bounded linear operator $P:H\to H$, $||P||=||P^*||$, where $||P||=\sup\{|(Pf,g)|:||f||\le 1,\ ||g||\le 1,\ f,g\in H\}$. Is it true that for any $f\in H$, ...
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### Show that if $||\cdot||_1$, $||\cdot||_2$ are equivalent norms then $(V,||\cdot||_1)$ is a banach space iff $(V,||\cdot||_2)$ is.

Show that if $||\cdot||_1$, $||\cdot||_2$ are equivalent norms then $(V,||\cdot||_1)$ is a banach space iff $(V,||\cdot||_2)$ is. I really didn't get it. Of course both spaces are normed spaces but ...