Question

How to understand ' Let $\mathcal{H}$ be a Hilbert space of functions $f$ : $ \mathcal{X} \rightarrow R$, denoted on a non-empty set $\mathcal{X}$.'

score 67 · Answer 1 · 2026-03-29 22:41:16

67

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How to understand ' Let $\mathcal{H}$ be a Hilbert space of functions $f$ : $ \mathcal{X} \rightarrow R$, denoted on a non-empty set $\mathcal{X}$.'

Published on 29 Mar 2026 - 22:41

score 87 · Answer 2 · 2026-03-29 22:42:20

87

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How to prove $\Phi_w(x)=\cos (w^T x+b)$ is outside the RKHS associated with the Gaussian kernel function$K(x,y)=\exp(-\frac{||x-y||^2}{2\sigma})$?

Published on 29 Mar 2026 - 22:42

score 115 · Answer 3 · 2026-03-29 22:43:34

115

Views

How to prove this specific kernel is not in RKHS?

Published on 29 Mar 2026 - 22:43

score 871 · Answer 4 · 2026-03-28 20:51:07

871

Views

Summary: Spectrum vs. Numerical Range

Published on 28 Mar 2026 - 20:51

score 741 · Answer 5 · 2014-12-29 16:25:05

741

Views

Why do bases of infinite dimensional spaces need to be orthonormal?

Published on 29 Dec 2014 - 16:25

score 3.2k · Answer 6 · 2026-03-25 22:25:58

3.2k

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Proof of the Riesz-Schauder Theorem (for compact operators) using the Analytical Fredholm Theorem

Published on 25 Mar 2026 - 22:25

score 1.1k · Answer 7 · 2026-03-29 05:12:23

1.1k

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Can Fourier transform be seen as a decomposition over a basis in a space of tempered distributions

Published on 29 Mar 2026 - 5:12

score 3.5k · Answer 8 · 2014-12-30 02:30:51

3.5k

Views

How to prove the sum of RKHS (Reproducing Kernel Hilbert Space)?

Published on 30 Dec 2014 - 2:30

score 5.6k · Answer 9 · 2014-12-31 18:35:24

5.6k

Views

A operator is unitary if and only if it is a surjective isometry

Published on 31 Dec 2014 - 18:35

score 970 · Answer 10 · 2026-03-27 00:05:03

970

Views

Existence of adjoint of the inverse

Published on 27 Mar 2026 - 0:05

score 56 · Answer 11 · 2026-03-27 06:06:43

56

Views

Is this extension a Hilbert space?

Published on 27 Mar 2026 - 6:06

score 41 · Answer 12 · 2019-10-02 21:54:51

41

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If $f(x) \in L^2(\mathbb R^d)$ then $f(x-y) \in L^2(\mathbb R^d \times \mathbb R^d)$

Published on 02 Oct 2019 - 21:54

score 259 · Answer 13 · 2019-10-03 10:06:56

259

Views

Using spectral theorem for integral Hilbert-Schmidt (maybe) operator

Published on 03 Oct 2019 - 10:06

score 50 · Answer 14 · 2026-03-25 19:06:03

50

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Norm of the solution of a Poisson equation

Published on 25 Mar 2026 - 19:06

score 133 · Answer 15 · 2026-03-27 00:10:21

133

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What is the square-root of the nonnegative self-adjoint operator $L^2\to L^2,f\mapsto\int f$?

Published on 27 Mar 2026 - 0:10

List Question

How to understand ' Let $\mathcal{H}$ be a Hilbert space of functions $f$ : $ \mathcal{X} \rightarrow R$, denoted on a non-empty set $\mathcal{X}$.'

How to prove $\Phi_w(x)=\cos (w^T x+b)$ is outside the RKHS associated with the Gaussian kernel function$K(x,y)=\exp(-\frac{||x-y||^2}{2\sigma})$?

How to prove this specific kernel is not in RKHS?

Summary: Spectrum vs. Numerical Range

Why do bases of infinite dimensional spaces need to be orthonormal?

Proof of the Riesz-Schauder Theorem (for compact operators) using the Analytical Fredholm Theorem

Can Fourier transform be seen as a decomposition over a basis in a space of tempered distributions

How to prove the sum of RKHS (Reproducing Kernel Hilbert Space)?

A operator is unitary if and only if it is a surjective isometry

Existence of adjoint of the inverse

Is this extension a Hilbert space?

If $f(x) \in L^2(\mathbb R^d)$ then $f(x-y) \in L^2(\mathbb R^d \times \mathbb R^d)$

Using spectral theorem for integral Hilbert-Schmidt (maybe) operator

Norm of the solution of a Poisson equation

What is the square-root of the nonnegative self-adjoint operator $L^2\to L^2,f\mapsto\int f$?

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