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

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2
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1answer
64 views

Unbounded extension of bounded operator

Is it possible to construct an unbounded extension of bounded densely defined operator? To be more concrete, let $\mathcal{H}$ be Hilbert space, $\mathcal{D}\subset\mathcal{H}$ - a dense subset, ...
0
votes
2answers
99 views

Parseval's identity, decomposition of inproduct.

Hoi, if $H$ is a seperable real Hilbertspace and $(e_n)$ orthonormal basis, then Parseval's identity $$\sum_n\left\langle x,e_n \right\rangle^2 = \left\|x\right\|^2 = \left\langle x,x ...
1
vote
0answers
53 views

Operators on a Hilbert space question

For a Borel measure $\mu$ define $\langle S_\mu x,y\rangle=\int_H\langle x,z\rangle \langle y,z\rangle \mu(z)$. An exercise in my book that I am reading says that I could find a $\mu$ s.t. $S_\mu$ ...
0
votes
1answer
59 views

Dense subspace in a Hilbert space

Let $H$ be a Hilbert Space and $\{e_n\}_{n\in\mathbb{N}}$ an orthonormal basis. Now let $(x_n)$ be a sequence in $H$ satisfying $$\sum_{n=1}^{\infty}||x_n-e_n||^2<1.$$ Prove that ...
0
votes
1answer
71 views

Common orthogonal basis for $L^2$ and $H^1$

How can we obtain a common orthogonal basis for the space $L^2(U)$ and $H^1(U)$ for some bounded open subset of $\mathbb{R}^n$? That this can be done is mentioned in Evans's Partial Differential ...
0
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1answer
58 views

Complex Projective Line

How can I go about showing that a collection of all states is the complex projective line $CP^1$? All I understand at the moment is that an element in $CP^1$ is of the form ...
1
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0answers
49 views

Closed affine subspace of $\mathcal{L}^{2}$

Consider a Hilbert space $\mathcal{L}^{2}=\lbrace X: X-\text{real-valued random variable}, \mathbb{E}(X^{2})<\infty \rbrace$ with the inner product $\langle X,Y\rangle=\mathbb{E}(XY)$. Let ...
2
votes
2answers
52 views

Show that $\ell^2(A)$ and $\ell^2(B)$ are isomorphic iff $A$ and $B$ have the same cardinality

Let $A,B$ be sets. Show that $\ell^2(A)$ and $\ell^2(B)$ are isomorphic iff $A$ and $B$ have the same cardinality. (Here $\ell^2(A)$ is the square integrable functions that stand on $A$ with the ...
0
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2answers
42 views

Adjoint of an operator question.

Let T be a normal operator. Prove that $\|T\|^{2n}=\|TT^*\|^n$ Has it got something to do with $\|T\|=\|T^*\|$?
0
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2answers
47 views

weak derivatives of exp(-|x|) and Hilbert Spaces

To which Hilbert Space (W^m,2) of R does the function exp(-|x|) belongs? I know its weak derivative is (-exp(-x) for x>0, exp(x) for x<0 and c0 (arbitrary) for x = 0). This weak derivative is in ...
0
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1answer
56 views

Show that H$(I)$ is a closed subspace of $L^2(I)$

EDIT: Original statement is not true, added condition. Let $I$ be the unit interval, define $H(I) = \{f\in AC(I)$ and $f'\in L^2(I)\}$. I want to show that $H(I)$ a closed subspace of $L^2(I)$. ...
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0answers
370 views

Is a closure of subspace N and and orthogonal complement of this subspace N orthogonal?

Ok, there is something I do not understand about what I run into today in an online document. I know it might sound simple but I am so new to topology so I am having hard time to understand. As we ...
0
votes
1answer
73 views

Counterexample of minimum principle in hilbert space on non closed but convex subspace

As I mentioned at title, I make tiny counterexample for minimum principle. Let $K=C([0,\frac{1}{2}]) \subset H=L^{2}([0,1])$. Then $K$ is convex since every $f,g \in K$, $(\alpha ...
2
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0answers
77 views

What's this standard duality argument?

I'm reading a proof of the Strichartz inequalities. It shows that $$ \| \int_\mathbb{R} e^{-is\Delta}F(s) \, ds \|_{L^2_x} \lesssim \|F\|_{L^{q'}_t L^{r'}_x}, $$ and then says that by duality, $$ ...
2
votes
1answer
91 views

On the completeness of inner product spaces.

Let $H$ be a Hilbert space, equipped with an inner product $(\cdot,\cdot)_1$ and norm $\|\cdot\|_1$ induced by it. Let $(\cdot,\cdot)_2$ be other inner product on $H$ and $\|\cdot\|_2$ the norm ...
1
vote
1answer
60 views

Condition for vector to be in the domain of unbounded operator.

Let $P$ be unbounded self-adjoint operator on some Hilbert space $\mathcal{H}$. We assume that the limit $$ \lim_{\epsilon \searrow 0} \|\exp(-\epsilon^2 P^2/2) P\psi\| $$ exists and is finite. Does ...
0
votes
1answer
68 views

Applying Schauder fixed point theorem to a map (explanation needed)

Let $F:L^2(\Omega) \to L^2(\Omega)$ be continuous map. Let $D$ be a function space. Since $F(L^2(\Omega)) \subset D$, and $D \subset L^2(\Omega)$ is a compact embedding, $F$ is a compact operator ...
4
votes
0answers
97 views

Ultraweak topology on Banach spaces

If $X$ and $Y$ are Banach spaces with $Y$ reflexive, then the space $\mathcal{B}(X,Y)$ of bounded operators from $X$ to $Y$ is the dual of the projective tensor product of $X$ and $Y^{*}$. As in the ...
0
votes
2answers
111 views

Spectral Theorem for normal operators

I want to prove this in the infinite dimensional Hilbert space case. What is the easiest way to go about this (What do I need to know, what theorems do I need,etc). My aim is to show every normal ...
0
votes
1answer
38 views

Question id a derivative on a Hilbert space

On a Hilbert space $H$; i have this function: $\tilde{f}(x)=f(x)+p(||x||)(x_0,x)$ where $x_0\in H, p\in C^2([0,\infty),\mathbb{R}),f\in C^2(H,\mathbb{R})$ i want to caculate $\tilde{f}', ...
1
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0answers
47 views

Continuity domain for momentum operator

The momentum operator in one dimension in quantum mechanics is $P=-i\frac{d}{dx}$ (with $\hbar=1$). Consider it as an operator on $L_2(0,2\pi)$, the space of square-integrable functions on $(0,2\pi)$. ...
3
votes
2answers
186 views

Derivative on Hilbert space

Please, on a Hilbert space what is the derivative of $\displaystyle\frac{x}{||x||}$ ? I know that it's equal to $\displaystyle \frac{1}{||x||}-\frac{\langle x,\cdot\rangle}{||x||^3} x$ but can I ...
1
vote
1answer
487 views

Supremum calculation

Calculate $\sup(\sum_{k=n+1}^{\infty}\frac{|x_{k}|^{2}}{4^{k} })$, where $x=(x_{1},x_{2},....)$ is a member of $l_{2}$ and the supremum is take over all $x$ with $||x||= 1$. My intuition says the ...
2
votes
1answer
53 views

Weak derivative of one parameter group and the domain of its generator

Let $U(t)=\exp(i t A)$ be a one parameter group generated by self-adjoint (unbounded) operator A. It is well-known that if $$ \lim_{t\rightarrow 0} \frac{U(t)\psi-\psi}{t} $$ exists then $\psi$ ...
1
vote
1answer
52 views

Distance between Unilateral shift and invertible operators.

I want to prove that the distance between unilateral shift and normal operators is $1$. But I need to prove that $d(S,\operatorname{Inv}(L(H))= 1$, where $H$ is a Hilbert space. Does anyone have any ...
2
votes
1answer
138 views

How to find adjoint operator?

Let $(X,\langle\cdot,\cdot\rangle)$ be a Hilbert Space over $K$ with orthonormal basis $(x_n)$, and let $(\lambda_n)\in K$ be a bounded sequence. The mapping $T:X\to X$ is defined by ...
1
vote
1answer
87 views

Generalized functions as integral kernels on Hilbert spaces

I'm a physics student and I'm studying functional analysis. I've got a doubt about some operators defined by integral kernels that are generalized functions. Let $L_2(a,b)$ be the Hilbert space of ...
1
vote
1answer
31 views

Logical meaning to morphisms between prehilbertian spaces

I was wondering how one can give a logical meaning to morphisms between prehilbertian spaces. If I was to consider such a morphism $f$ as a logical morphism between two $L$-structures, I should have ...
6
votes
1answer
51 views

Derive Fourier transform from what it should do?

I was wondering about the following: Imagine you want to figure out whether there is a transform that exchanges differentiation with multiplication and convoution with pairwise transformation for ...
2
votes
1answer
150 views

The trace class operators are the dual of the compact operators

I know that the map from the trace class operators $L_1(H)$ to the dual of the compact operators $K'(H)$ given by $A \mapsto tr( \cdot A)$ is an isometric isomorphism. Linearity is obvious by the ...
0
votes
2answers
60 views

Orthogonal Projections in $U\subset V$ two subspaces in a Hilbert space

Let $U,V\subset H$ be closed subspaces of a Hilbert space $H$, and let $P_U$ and $P_V$ the respective orthogonal projections. Show: $U\subset V \Longleftrightarrow P_U=P_UP_V=P_VP_U$ Trying to ...
1
vote
1answer
46 views

Can somebody explain the well-defined mapping to me?

Let $(X,(\langle.,.\rangle)$ be a Hilbert space over $\mathbb{K}$ with an orthonormal basis $(x_n)_{x\in\mathbb{N}}$ and let $(\lambda_n){n\in\mathbb{N}}\subset \mathbb{K}$ be a bounded sequence.The ...
2
votes
2answers
121 views

Is $H^2\cap H_0^1$ equipped with the norm $\|f'\|_{L^2}$ complete?

Let $-\infty<a<b<+\infty$. Consider the norms $\|\cdot\|_{L^2}$, $\|\cdot\|_0$ and $\|\cdot\|_{H^1}$ defined on suitable spaces and given by ...
1
vote
2answers
74 views

Chart of how the mathematical spaces are related? (soft question)

When dealing with specific function spaces e.g. Sobolev, Hilbert, etc., I find it easy enough to accept the properties of that space and work with them; however, I have a hard time visualizing how ...
4
votes
1answer
42 views

Is this a correct argument for why this space of sequences is a Hilbert space?

I am assigned the following problem (a piece of Exercise 6.5 in Brezis's book): Let $(\lambda_n)$ be a sequence of positive numbers such that $\lim_{n\to\infty}\lambda_n=+\infty$. Let $V$ be the ...
1
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1answer
191 views

The Trace Class Operators Form a Banach Space

I want examining the trace class operators $L_1(H)$ of a separable Hilbert space $H$ with the norm $||A||_1=\sum\limits^{\infty}_{n=1}\lambda_n$ where $\lambda_n$ are the eigenvalues of ...
3
votes
2answers
251 views

Linear operators with no adjoint

Here is a standard theorem about bounded operators: Let $H$ be a Hilbert space. For any bounded linear operator $A:H\to H$ there is a unique bounded operator $A^*$ s.t $\langle Au,v\rangle=\langle ...
5
votes
3answers
161 views

Gram-Schmidt for uncountable sets?

I know that Gram Schmidt can be applied to countable linear independent sets on Hilbert spaces, but what happens if we apply it on uncountable sets? Obviously this process has to fail then (at least ...
0
votes
0answers
24 views

Inner product spaces, examples for subspaces with certain properties [duplicate]

Let $H$ be a inner product space. Give examples for a subset $U\subset H$ so that (a) $\overline{U}\neq U^{\bot\bot}$ (b) $\overline{U}\oplus U^{\bot}\neq H$ I have thought about $H=C[0,1]$ with ...
5
votes
1answer
140 views

Adjoint of multiplication by $z$ in the Bergman space

I am learning Hilbert space theory from Halmos' "Introduction to Hilbert space and the theory of spectral multiplicity". While talking about understanding adjoints (p. 39), he calls special ...
0
votes
1answer
147 views

Proof Riesz representation theorem

I have a question regarding the proof of the Riesz representation theorem. Why do we declare the isomorphism $\Phi: H \rightarrow H'$ in an antilinear way? I mean if, this isomorphism would pick the ...
1
vote
1answer
159 views

Weak convergence equal to coordinate-wise convergence

Show for the Hilbert Space $\ell^2(\mathbb{N})$ weak convergence of a bounded sequence $(x_k)_{k \in \mathbb{N}}$ is equal to coordinate-wise convergence. A sequence $(x_k)$ is weak ...
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0answers
41 views

Continuity of certain projections in the weak topology.

I'd like to prove that: Given a Hilbert space H and S a closed subespace, $S \subseteq H$, the projection $P_{S}:H \to S$ is continuous in the weak topology. I have tried the following. ...
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0answers
34 views

$C_0$-Semigoups - References needed

I'm looking for the references which contains the following theorem with a proof: Theorem Let $(T_t)$ be a uniformly continuous semigroups with generator $A \in B(H)$, where $H$ is a Hilbert space. ...
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2answers
90 views

Self-adjoint Hilbert Space operators

Let $H$ be a Hilbert space and $T$ is a self adjoint continuous operator in $\mathcal B(H)$. Show that $\|T^{2^{k}}\| = \|T\|^{2^{k}}$. Does this equality hold for all operators? Now it is clear that ...
3
votes
1answer
671 views

In a Hilbert space, every bounded and closed set is weakly relatively compact.

My aim is to prove that in a Hilbert space, any sequence has a weakly convergent subsequence. To prove this, I'm trying to prove that: ...
2
votes
1answer
88 views

Superspace as the Hilbert Space for Quantum Gravity

This is a question I've asked in physics.stackexchange, but have obtained no answers: Let $\mathcal{A}$ be the Ashtekar connection. Since $^{(3)}g_{AB}=i\frac{\delta}{\delta\mathcal{A}^{AB}}$ (see R. ...
0
votes
1answer
68 views

Convergent series in Hilbert space

I am looking for a proof of the following theorem. Consider a countable orthonormal set in Hilbert space $H :\ \ u_1, u_2, ...$ $\sum_{j=1} ^{\infty} r_ju_j$ is convergent in $H \iff \sum_{j=1} ...
0
votes
1answer
30 views

Check this is a hilbert norm: $ \ell^2 $ with norm $\| \cdot \| := \| \cdot \|_{\ell^2} + \| \cdot \|_{\ell^p}$

Clearly $ p \geq 2 $ so it gains sense calculating the $\ell^p $-norm. According to my calculation this norm is equivalent to the $\ell^2 $ norm, in fact given a cauchy sequence w.r.t $\| \cdot \| $ ...
2
votes
1answer
84 views

Hilbert space, orthonormal system, compact set of vectors

Could you help me solve this problem? Let $e_1, e_2, ...$ be an orthonormal system in a Hilbert space, $\delta_1, \delta_2 ... \in (0, + \infty)$. Prove that the set of all vectors $\sum _{n=1} ...