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

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8
votes
1answer
131 views

How to Prove the Semi-parametric Representer Theorem

This question concerns the generalized Representer Theorem, due to Schölkopf, Herbrich, and Smola. In this magnificent work, the authors provide two versions of the Representer Theorem, a ...
7
votes
1answer
483 views

Isometric to Dual implies Hilbertable?

Let $X$ be a Banach space and suppose that $X$ is isometric to its continuous dual space $X^*$. Must $X$ be hilbertable in the sense that there exists an inner product which induces the norm on $X$? ...
7
votes
3answers
2k views

A linear operator on a finite dimensional Hilbert space is continuous

How do I show that a linear function from a Hilbert space $H$ to itself is continuous if $H$ is finite dimensional? Also, what would be an example of a linear function from a Hilbert space to itself ...
6
votes
3answers
259 views

A complete orthonormal system contained in a dense sub-space.

Let H be a separable complex Hilbert space. Let A be a dense sub-space of H. Is it possible to find a complete orthonormal system for H that is contained in A?
6
votes
2answers
2k views

Weak Convergence implies boundedness and componentwise convergence

Let $\ell^2$ be the set of real number sequences $\{a_n\}$ such that $\sum a_n^2 <\infty$. Let $\langle a_n,y\rangle \rightarrow \langle a,y\rangle$ for some $a\in \ell^2$ and for all $y\in ...
6
votes
1answer
246 views

Two different definitions of ellipticity

This is a question originating in another mathematics forum, matematicamente.it (in Italian). In literature one encounters the word elliptic in (at least) two different definitions. In what follows ...
5
votes
3answers
189 views

Sequences are Cauchy depending on the norm?

Assume that we have two Banach spaces $B_1, B_2$ with their underlying sets being identical. Is it possible that a Cauchy sequence in one of the spaces would fail to be Cauchy in the other? I am ...
2
votes
1answer
118 views

if $E^2=E$, then $\text{Im}\;E\subset\left(\ker E+(\ker E)^\perp\right)$?

Notation: $V$ is a infinite-dimensional inner product space; $\langle\cdot,\cdot\rangle$ is the inner product of $V$; $E:V\rightarrow V$ is a linear map; $\text{Im}=\{E(v):v\in V\}$; $\ker E=\{z\in ...
7
votes
2answers
247 views

Counterexample for the stability of orthogonal projections

Let $V$ be a seperable Banach space, which is dense and continuously embedded in a Hilbert Space $H$. Let $(V_m)$ be a Galerkin scheme (See definition below) for $V$. Using the embedding we can ...
4
votes
0answers
114 views

Is this projection operator onto a subspace of a Hilbert space bounded?

(I copy and paste and edit from Is this operator bounded? Hilbert space projection, my question is almost the same) Let $V \subset H$ be Hilbert spaces (different inner products) with $V$ dense and ...
3
votes
4answers
1k views

Orthogonal complement of a Hilbert Space

I have this problem: Let $S$ be a subset of a Hilbert $H$ and let $M$ be the closed subspace generated by $S$. Show that $M^{\perp} = S^{\perp}$ $M = (S^{\perp})^{\perp}$ if $V$ is a subspace of ...
3
votes
1answer
359 views

Isomorphic Hilbert spaces

As part of a broader proof , I need to show that every two separable Hilbert spaces (that contains a dense countable set) are isomorphic (the linear mapping from one space to the other is injective ...
3
votes
2answers
641 views

$\ell_{p}$ space is not Hilbert for any norm if $p\neq 2$

My question is motivated by this one: $\ell_p$ is Hilbert space if and only if $p=2$ Maybe it is a simple thing or im just confused but, suppose we are given any norm in $\ell_{p}$ for $p\neq 2$. ...
2
votes
0answers
128 views

Projection and Pseudocontraction on Hilbert space

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let ...
2
votes
1answer
202 views

Compact operator on $l^2$

Let A be a bounded linear operator on $l^2$ defined by A($a_n$)=($\frac{1}{n} a_n$). Would you help me to prove that A is compact operator. I guess the answer using an approximation by a sequences of ...
1
vote
1answer
45 views

In a dagger category, is there a name for morphisms $f : X \rightarrow Y$ with $\mathrm{id}_X = f^\dagger \circ f$?

In a dagger category, is there a name for morphisms $f : X \rightarrow Y$ with $\mathrm{id}_X = f^\dagger \circ f$? Clearly, every such arrow is a split monomorphism; further, if such an $f$ is ...
1
vote
2answers
178 views

Hilbert dual space (inequality and reflexivity)

Let $V \subset H$ where $H$ is Hilbert space. Let $T:H^* \to V^*$ be the canonical map that restricts the domain of a functional in $H$ so that it's a functional in $V$. How do I show that $$\lVert ...
0
votes
1answer
56 views

Find minimal $\alpha_3$ such that $u\in H^3(\Omega)$ and $u(x,y,z)=x^\alpha(1-x)y(1-y)z(1-z)$

My instructor presented me the quiz below but forgot to define key terms such as minimality and $H^3$. Quiz Let $u(x,y,z)=x^\alpha(1-x)y(1-y)z(1-z)$. Find the minimal $\alpha_3$ such that $u\in ...
7
votes
1answer
224 views

Energy estimate of the differential equation $\dot{x}=Ax$

Conside the differential equation $$\dot{x}=Ax,\qquad x(t):{\bf R}\to{\mathcal H}$$ where $\mathcal{H}$ is a Hilbert space and $A$ is a bounded linear operator. With the initial condition, one can ...
5
votes
2answers
228 views

Why is the numerical range of a self-adjoint operator an interval?

I was reviewing for a test for functional analysis when I came across the following statement: Let $T$ be a bounded self-adjoint operator on a Hilbert space $H$. Then the numerical range of it is ...
5
votes
2answers
885 views

Elegant proof that $L^2([a,b])$ is separable

Is anybody aware of, or can provide at least an outline, of a proof that the Hilbert space of Lebesgue functions square-integrable on the closed real interval [a,b], equipped with the $L^2$ norm, is ...
4
votes
2answers
158 views

Gram-Schmidt in Hilbert space?

EDIT: After some contemplation I decided to phrase the question better to avoid trivial answers. Consider a Hilbert space with a basis $\{v_{i}\}$ where $i\in I$ an index set, which could be ...
4
votes
2answers
179 views

Is this operator bounded? Hilbert space projection

Let $V \subset H$ be Hilbert spaces (different inner products) with $V$ dense in $H$. Let $b_n$ be an orthonormal basis for $H$ and an orthogonal basis for $V$. Define $$P_n:H \to ...
3
votes
1answer
170 views

The unit ball in a Hilbert space

I have a request for any ideas to prove: If $H$ is a Hilbert space, then any unit vector is an extreme point of the unit ball of $H$. Every isometry is an extreme point of the unit ball of the ...
3
votes
2answers
55 views

Question about finding minimum-Hilbert spaces

How to find $$\min_{a,b,c\in\mathbb{C}}{\int_0^{\infty}} |a+bx+cx^2+x^3|^2 e^{-x} dx = ?$$ Thanks in advance.
2
votes
0answers
87 views

The support of Gaussian measure in Hilbert Space $L^2(S^1)$ with covariance $(1-\Delta)^{-1}$

Let $\mu$ be Gaussian measure defined on Hilbert space $\mathcal{H}=L^2(S^1)$ ($S^1$ - circle) by formula $$ \int e^{(f,g)} d\mu(f) = e^{-\tfrac{1}{2}(g,C g) }. $$ The covariance operator $C$ is ...
2
votes
1answer
332 views

Derivative of Convex Functional

Suppose that $H$ is a real Hilbert space and that $f:H \to \mathbb{R}$ is differentiable in the Frechet sense. Then we can think of the derivative as a function $f': H \to H^* = H$. Suppose that this ...
2
votes
3answers
389 views

$C[0,1]$ is NOT a Banach Space w.r.t $\|\cdot\|_2$

I'm trying to find a cauchy sequence in $C[0,1]$ that converges under $\|\cdot\|_2$ to a limit which isn't continuous. Any ideas?
2
votes
2answers
2k views

A proof of the Riesz representation theorem

I'm having trouble filling the steps in this guided proof of Riesz's representation theorem. (I already have a proof I can understand, but I'd like to understand this one too.) Let $H$ be a Hilbert ...
2
votes
3answers
634 views

Orthonormal basis in $L^2(\Omega)$

In the one dimension case, where $\Omega\subseteq{\bf R}$ is a bounded domain, for example $\Omega=[0,2\pi]$, one can find the orthonormal basis for $L^2(\Omega)$: $$\{e_n\}_{n\in {\bf Z}}$$ where ...
1
vote
2answers
76 views

Isometry <=> Adjoint left inverse [duplicate]

Is it true that: $$T\text{ isometric}\iff T^*\text{ left inverse}$$ Obviously: $$\text{"}\Rightarrow\text{": }\langle x,\tilde{x}\rangle=\langle Tx,T\tilde{x}\rangle=\langle x,T^*T\tilde{x}\rangle$$ ...
1
vote
1answer
56 views

Why is $\langle f, u \rangle_{H^{-1}, H^1} = (f,u)_{L^2}$ when $f\in L^2 \cap H^1$ and not $\langle f, u \rangle_{H^{-1}, H^1}=(f,u)_{H^1}$?

More generally, if $V \subset H \subset V'$ are Hilbert spaces, why is $$\langle f, u \rangle_{V',V} = (f,u)_{H}$$ when $f\in H \cap V$ and not $$\langle f, u \rangle_{V',V}=(f,u)_{V}?$$ Is this what ...
1
vote
2answers
761 views

Hilbert Space is reflexive

A normed space $X$ is reflexive iff $X^{**}=\{g_x:x\in X\}$ where $g_x$ is bounded linear functional on $X^*$ defined by $g_x(f)=f(x)$ for any $f\in X^*$. Let $X$ be a Hilbert space, would you help ...
1
vote
1answer
213 views

Unique extension to a bounded operator

Suppose $\left\{ e_{1},e_{2},\ldots\right\} $ is an orthonormal basis for a Hilbert space $\mathcal{H}$ and for each $n$ there is a vector $Ae_{n}$ in $\mathcal{H}$ such that $\sum\left\Vert ...
1
vote
2answers
325 views

Norm of the sum of projection operators

Is it true that $$|| a R+b P||\leq\max \{|a|,|b|\},$$where $a$ and $b$ are complex numbers and $P,R$ are (orthogonal) projection operators on finite-dimensional closed subspaces of an ...
1
vote
3answers
1k views

Question on weak convergence ( Example).

Can anybody tell me why $\sin(nx)$ converges weakly in $L^2(-\pi,\pi)$. I can't see how $\sin(nx)$ can converge? Explanation with any other example will be nice as well.
1
vote
3answers
421 views

minimum principle in Hilbert space

Minimum principle is following: Let $M$ be a closed convex nonempty subset of Hilbert space. Then there exists $x\in M$ which have a minimum norm. Assume that $M$ is not convex subset. What is a ...
0
votes
2answers
232 views

Finding the min of an integral

So I have to find the following $$\min_{a,b,c\in\mathbb{R}}\int_{-1}^{1} |x^3-a-bx-cx^2|^2dx$$ I have a hint at a solution which says to consider $X=\{\mbox{polynomials of degree} \leq 2\}$. So then ...
0
votes
1answer
148 views

Averaging differential forms

Let $M$ be a manifold with a circle action, i.e. with a 1-parameter group of diffeomorphisms $\phi_t:M\to M$ of period 1. I think of the average of a differential form $\omega \in \Omega^n(M)$ with ...
6
votes
1answer
224 views

Approximating a Hilbert-Schmidt operator

Let $H$ be a separable Hilbert space. Recall that a bounded operator $A : H \to H$ is said to be Hilbert-Schmidt if $$\|A\|_{HS}^2 := \sum_{i=1}^\infty \|A e_i\|^2 < \infty$$ where ...
6
votes
2answers
312 views

Question about positive operators on a Hilbert space

I have the following problem. Let $\Omega \subset R^n$ have finite measure, let $H = L^2(\Omega)$ and let $S: H \to H$ be a bounded linear operator. Then it is well known that $P = SS^*$ is a positive ...
5
votes
1answer
68 views

Nonseparable $L^2$ space built on a sigma finite measure space

Is it possible to have a nonseparable $L^2$ Hilbert space for which the underlying measure space is sigma finite? I appreciate any example but prefer one built on the Borel sigma algebra of some ...
5
votes
1answer
406 views

Using Lax Milgram to find a weak solution in an intersection of Sobolev spaces

I am trying to prove the existence of a weak solution of the problem: $$ -\Delta^2 u = f \in L^2(U)\\ \\ u|_{\partial U}=\Delta u|_{\partial U} = 0 $$ on the bounded open set $U\subset\mathbb{R}^n$ ...
5
votes
1answer
154 views

Show $T$ is compact

$H$ and $K$ are Hilbert Spaces, $(u_n)$ and $(v_n)$ are sequences in $H$ and $K$ respectively. $\sum_{n=1}^{n=\infty} \|u_n\|\|v_n\| $ converges. $T\colon H\rightarrow K$ is defined by ...
5
votes
2answers
341 views

Orthonormal basis for Sobolev Spaces

Sobolev spaces of order 2 are known to form a Hilbert space. Consider such a Sobolev space of (order 2) functions on the domain $f:\mathbb{R}\rightarrow \mathbb{R}$. What is an example for the basis ...
5
votes
2answers
773 views

Relationship of Fourier series and Hilbert spaces?

I just read in a textbook that a Hilbert space can be defined or represented by an appropriate Fourier series. How might that be? Is it because a Fourier series is an infinite series that adequately ...
5
votes
2answers
541 views

Recognition of an orthonormal complete set in an Hilbert Space

I recently came across this exercise: Let $u_n:[0,1]\to \mathbb R$ be the sequence of functions defined by: $$(u_n):=\text{sign }(\sin(2^n\pi x)),\qquad n=0,1,2,\dots$$ a)Prove that this set of ...
4
votes
1answer
93 views

Question about adjoint map and strong operator topology (SOT)

I am wondering if there is any condition one can apply (e.g. uniform boundedness?) that ensure the adjoint of a net of SOT-continuous elements is again SOT-continuous? My major question is ...
4
votes
1answer
171 views

Normal $T\in B(H)$ has a nontrivial invariant subspace

I am wondering if the following is true: Every normal $T\in B(H)$ has a nontrivial invariant subspace if $\dim(H)>1$?
4
votes
1answer
238 views

Showing the basis of a Hilbert Space have the same cardinality

I am trying to show that if we have two orthnormal families, $\{a_i\}_{i\in K}$ and $\{b_j\}_{j\in S}$ and these are the basis of some Hilbert Space H then they have the same carnality. So If I ...