fa.functional analysis – Extension of a Hilbert basis

The picture below is taken from this paper: http://real.mtak.hu/22877/.

The authors claim that the basis of $H^2(Omega) cap H^1(Omega)$ denoted by $lbrace w_i rbrace _{i geq 1}$ can be extended to be a basis of $L^2(Omega;H^1(0,1))$. I don’t see how it can be possible. In my thinking, we have to multiply $w_i(x)$ by $h_i(x)$ where $h_i(x)=frac{cos(i pi x
)}{ipi}$
is a basis of $H^1(0,1)$. Is this write?. Thank you.

enter image description here

propositional calculus – Reference for proof of completeness of classical predicate logic in a Hilbert system

Many proofs of the completeness of classical logic with respect to some particular Hilbert style atomization of it do not explicitly reference the axioms at hand. The devils must be buried in the details, and I am looking for a proof that shows how each axiom contributes to completeness.

For example, intuitionistic propositional calculus can be axiomatized as classical logic minus the law of excluded middle. Therefore, any completeness proof of classical logic must reference the axiom of the law of excluded middle, else such a proof can’t distinguish between classical logic and intuitionistic logic.

The Henkin style proofs I have seen are rather abstract and don’t reference the specific axiom system at hand.

Can someone please offer a reference that does include the gritty details involving all the axioms?

fa.functional analysis – Operator targeting $ell^2$-direct sum of Hilbert spaces continuous if all its projections are continuous?

Let $(U, |cdot|_U)$ be a Banach space, $(H_k, |cdot|_k)_{kinmathbb{N}}$ be a sequence of Hilbert spaces and denote by

$$tag{1}H:=bigoplus_{k=1}^infty H_k equiv left{h=(h_k) middle| h_k in H_k, ,forall kinmathbb{N} quad text{and} quad |h|_H^2:=sum_{k=1}^infty|h_k|_k^2 < infty right}$$

the $ell^2$-direct sum of these spaces (which is known to be a Hilbert space itself).

Let further $pi_k : H rightarrow H_k$, $pi_k((h_k)) := h_k$, be the projection of $H$ onto its $k^{mathrm{th}}$-component.

Question: Is it true, then, that a (not necessarily linear) map $T : U rightarrow H$ is $textit{continuous}$ if its projections

$$tag{2}T_n := pi_kcirc T quad text{are continuous} quad text{for each } kinmathbb{N}?$$

Remark: If necessary, it may be assumed that each of the $H_k$ are finite-dimensional.

Any references, hints or proofs (or indeed counterexamples) that cover this are appreciated!

Is t there a nontrivial compact Hilbert space?

Is there a nontrivial compact Hilbert space?

Hilbert Transform Calculation of a pulse

Can anybody help me with the Hilbert transform of the following pulse:
$$
p(t) = frac{8Brho cos(2pi Bt+2pi Brho t)+frac{sin(2pi Bt-2pi Brho t)}{t}}{pisqrt{2B}(1-64(Brho t)^2)}
$$

I tried by convolution of the above signal with $h(t) = 1/pi t$ but I am not able to simplify it.
Can anyone suggest an approach?

fractals – Hemispherical space filling hilbert curve

first question here, sorry for any posting infractions.

I need to create/find/buy a hemispherical space-filling Hilbert(or similiar) curve.
something similiar to Cube hilbert
but only filling a hemisphere.

And this might be immposible but I would like to keep spacing between lines even as they approach the center…

Would appericiate any info pointing me in the right direction.

orthogonality – In a Hilbert Space: Minimal $Leftrightarrow$ Biorthogonal

Perhaps obvious, but I’m struggling a bit with (i) $Rightarrow$ (ii) only.

Proposition. Let ${x_n}_{n in mathbb{N}}$ be a sequence in a Hilbert space $H$. Then the following are equivalent:

i) Let $S_n := overline{text{span}{x_n : n neq m}}$. For each $m in mathbb{N}$, $x_m notin S_n$.

ii) Exists ${y_n}_{n in mathbb{N}}$ in $H$ that is biorthogonal to $x_m$, i.e. the inner product is $1$ when $n = m$ and $0$ otherwise.


My idea: I suppose if we let $S_n^{perp}$ be the orthogonal complement of $S_n$, then for any $y_n in S_n^{perp}$, we have $langle x_m, y_n rangle = 0$. But when $n = m$, how can I be sure that $langle x_m, y_n rangle = 1$?

oc.optimization and control – Proximal operator of composition with norm squared on a Hilbert space

Let $(X,Sigma,mu)$ be a finite measure space and let $L^2(mu,mathbb{R}^d)$ denote the Bochner space of strongly measurable functions taking values in $mathbb{R}^d$. Let
$$
begin{aligned}
T &:L^2(mu,mathbb{R}^d)rightarrow L^2(mu,mathbb{R})\
& fmapsto |f|_2,
end{aligned}
$$

where $|cdot|_2$ is the Euclidean norm on $mathbb{R}^d$. Fix a $phiinGamma_0(L^2(mu,mathbb{R}))$.

Is there a known expression for
$
operatorname{Prox}_{Tcirc phi}
$

in terms of $operatorname{Prox}_{phi}$, $phi$, and $T$?

functional analysis – How to prove that $H^2_{l_0}$ is a Hilbert space.

Let $I=(l_0,l_1)subset mathbb{R}$ and consider the space

$$H^2_{l_0}={uin H^2(I): u(l_0)=u'(l_0)=0},$$
endowed with the inner product $left<u,vright>_{H^2_{l_0}}:=left<u”,v”right>_{L^2(I)}=int_I u”v”,dx$, for real valued functions.

Here $H^2(I):=W^{2,2}(I)$

I have proven that $left<cdot,cdotright>_{H^2_{l_0}}$ defines an inner product, it also defines a norm. So far I have seen $H^2_{l_0}$ is a pre-Hilbert space, but I’m having troubles proving the completeness with the metric induced by the inner product.

I really appreciate your help.

fa.functional analysis – Interpolation of embedded Hilbert spaces and intersection

I’m wondering under what hypothesis it is true a property like

$$(mathcal{H}_1cap X, mathcal{H}_2cap X)_{theta}=mathcal{H}_1cap Xcap (mathcal{H}_1, mathcal{H_2})_{theta}$$

where $mathcal{H}_2hookrightarrow mathcal{H}_1$ are Hilbert spaces contained in a larger Hilbert space $mathcal{H}$ with $Xsubset mathcal{H}$.

I’m not skilled in interpolation theory, but here is my attemp. In Triebel’s book Section 1.17.1 (https://www.sciencedirect.com/bookseries/north-holland-mathematical-library/vol/18) there is a Theorem which read as follows

Theorem 1: Let ${A_0, A_1}$ be an interpolation couple. Let $B$ be a complemented subspace of $A_0+A_1$ whose projection belongs to $L({A_0, A_1}, {A_0, A_1})$. Let $F$ be an arbitrary interpolation functor. Then ${A_0cap B, A_1cap B}$ is also an interpolation couple and
$$F({A_0cap B, A_1cap B})=F({A_0, A_1})cap B$$

In my case, the interpolation couple would be ${mathcal{H}_2, mathcal{H}_1}$ and $B=mathcal{H}_1cap X$. If $X$ is such that $mathcal{H}_1cap X$ is a closed subspace of $mathcal{H_1}$ then it is also a complemented subspace of $mathcal{H}_1$ whose projection is linear continuous in $mathcal{H}_1$ (i.e. belongs to $L(mathcal{H}_1)$). The previous reasoning implies that I’m able to apply the previous Theorem to arrive my initial statement, or I’m missing something?

I know that interpolation is not well behaved with respect to restriction (https://math.stackexchange.com/questions/3542640/complex-interpolation-and-intersection) and I didn’t find much more results than the previous one in the literature. Every hint or reference is very well received!

Remark: I asked in some generality, but I’m treating a particular case where $mathcal{H_2}, mathcal{H_1}$ are sobolev spaces $H^k$, the larger space is $L^2$ and $X$ is the domain of a maximal monotone operator, in some interval $(0, L)$.