# Questions tagged [hilbert-spaces]

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

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### Interpreting the Definition of Gaussian White Noise

My book defines Gaussian white noise as an isometry from $L^2(X,\mathscr{F}, \mu)$ to a vector space of Gaussian random variables with mean zero. Can someone explain what on earth this isometry has ...
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### On the convergence of approximate units for C*-algebras.

Let $A$ be a non-unital C*-algebra and let $\pi : A \to \mathcal{B}(H)$ be a non-degenerate representation of $A$ (that is, $\mathrm{ span }\{\pi(a)h : a \in A, h \in H\}$ is a dense subset of $H$). ...
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### An orthonormal basis of $L^{2}(\mathbb{R}^{n})$ which is pointwise in $\ell^{2}(\mathbb{N})$?

Does there exist an orthonormal basis of the Hilbert space $L^{2}(\mathbb{R}^{d})$, say $(e_{n})_{n=1}^{\infty}$, such that all elements $e_n\in L^{2}(\mathbb{R}^{d})\cap C^{0}(\mathbb{R}^{d})$ and ...
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### How to obtain the feature map from this kernel function

Here in this paper: https://ideas.repec.org/a/eee/ejores/v292y2021i3p1004-1018.html , the author has written that $\phi (\cdot )$ represents the feature map from input space $\mathcal{X}$ to the high ...
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### Nets and closedness of $\mathcal{B}(H)$ in the topology of pointwise convergence

Let $H$ be an infinite-dimensional Hilbert space and $\mathcal{B}(H)$ the set of bounded linear operators on $H$. One way to define the strong operator topology (SOT) on $\mathcal{B}(H)$ is by ...
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### Is $A\in B\left( \overline{H}\right)$ invertible if it is onto and $\left. A\right\vert _{H}\in B\left( H\right)$ is one-to-one and onto?

Let $\overline{H}$ be a Hilbert space and $A\in B\left( \overline{H}\right)$ be a surjective operator such that $A\left( H\right) =H$ and its restriction to $H$ is injective and. I'm asking if under ...
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### Why $H^1(0,1) = L^2(0,1)$ is not good?

From the theory of Hilbert bases and separability considerations one can easily derive a Hilbert space isomorphism between $H^1(0,1)$ and $L^2(0,1)$. I am convinced that such an isomorphism is useless:...
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### Inner products and unbounded linear operators in Hilbert spaces

Suppose I have four compact, injective linear operators on a Hilbert space $A$, $B$, $C$, and $D$ and two elements of the Hilbert space $a$ and $b$. We know that $AB^{\star}=CD^{\star}$, $C[a]$ is in ...
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### Hilbert Schmidt embedding (needed for Cylindrical Wiener processes)

In Karczewska's paper (2005) it says: "Assume that $U_1$ is an arbitrary Hilbert space such that $U$ is continuously embedded into $U_1$ and the embedding of $U_0$ into $U_1$ is a Hilbert-Schmidt ...
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### If ${x_n}$ is weakly convergent to $x$ in $L^p(0,1)$ space, and if $\lim_{n}\|x_n\|=\|x\|$, then if $\lim_{n}x_n=x$? [duplicate]

If ${x_n}$ is weakly convergent to $x$ in $L^p(0,1)$ space, and if $\lim_{n}\|x_n\|=\|x\|$, then if $\lim_{n}x_n=x$? This question is from problem 6.1.4 in Avner Friedman's "Foundations of ...
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### Rudin's RCA Theorem $4.18$.

There is the definition which we need for the proof: There is the theorem which we need for the $4.18$: There is $4.18$: Let {$u_\alpha : \alpha$ $\in$ $A$} be an orthonormal set in $H$. Each of ...
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### Bounded operator with finite continuous spectrum

Let $T$ be a bounded operator on a complex Hilbert space $H$. Let $\sigma_c(T)$ denote its continuous spectrum. It seems that $\sigma_c$ can in general be quite weird and not "continuous" in ...
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### If $A,B\subset H$ are closed subspaces and $\operatorname{codim}(A+B)<\infty$ do we have $A+B$ closed?

Let $H$ be a Hilbert space and $A,B\subset H$ be two closed subspaces such that $\operatorname{codim}(A+B)<\infty$. I would be very surprised if it tuns out that $A+B$ is not necessarily closed. I'...
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### Proof of Convergence of the Sum of Components in a Hilbert Space

I recently started studying the fascinating mathematical structures of Hilbert spaces. As a physics guy, I worked with Hilbert spaces in quantum mechanics without knowing the rigorous definition of ...
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### Proving Equality Regarding the Adjoint of Bounded Linear Operator

I am proving the Proposition 2.13 from Elementary Functional Analysis by MacCluer, mainly on c) and d) We're given that For any $A,B \in \mathscr{B}(\mathscr{H})$, we have \begin{align*} (\alpha A)^* &...
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### The proof of $||T|| = \sup\{|(T(f),f)| \;|\; ||f|| = 1 \}$ when $T = T^*$

I'm studying a property of symmetric linear operator in Hilbert space in Stein's Real analysis chapter 4. When $T = T^*$, then $\Vert T\Vert = \sup\{|(T(f),f)| \;|\; \Vert f\Vert = 1 \}$ The following ...
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### linear functional and inner product

Currently, I'm studying the relationship btw linear functional and inner product, and this is the associated theorem in Stein's Real Analysis. (theorem 5.3, chapter 4) I understand all of the proofs, ...
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### Spectrum of a $T^*T$ for $T$ normal

Let $T$ be a bounded normal operator on a complex Hilbert space $V$. Suppose $\alpha \in \mathbb{C}$ is in the spectrum of $T$, i.e. $T - \alpha I$ is not invertible. I'd like to show that $|\alpha|^2$...
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