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

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Show these operators converge to a particular limit

Let $H$ be a Hilbert space, and $T$ be a operator on $H$ of the form $T=\sum_{n=1}^{\infty}{\lambda}_{n}<x,e_{n}>e_{n}$ where $e_{n}$ are the eigenvectors of $T$ and an orthonormal basis of H ...
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1answer
70 views

Naturality of Riesz' Representation

What does it mean precisely in the context of category theory when somebody says that Riesz' representation is canonic resp. every Hilbert space is naturally antiisomorphic to its dual?
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1answer
75 views

A question about closed balls in Hilbert Space.

Let $H$ be a separable and infinite dimensional Hilbert Space and let $B$ be a closed ball of $H$ whose diameter is some positive real number. Is every covering of $B$ by closed bounded subsets of ...
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1answer
18 views

If $u_n \rightharpoonup u$ and $Au_n \rightharpoonup b$, does it follow that $\limsup_{n \to \infty}(Au_n, u_n) \leq (b, u)$ if $A$ is continuous?

Let $H$ be a Hilbert space and let $A:H \to H$ be a (nonlinear) continuous map. If $u_n \rightharpoonup u$ and $Au_n \rightharpoonup b$, does it follow that $$\limsup_{n \to \infty}(Au_n, u_n) \leq ...
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366 views

Duality mappings on finite-dimensional spaces

I have a few questions regarding some concepts in a book "nonlinear partial differential equations by Roubicek" I am studying. The following is from the text. "Let $V$ be a separable, reflexive ...
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94 views

Product of strong and weakly converging sequences

Consider a sequence of functions $\{u_n\}\in L^2([0,T],L^2(\Omega)) $ which converges strongly to a function $u\in L^2([0,T],L^2(\Omega))$. Then $u_n \rightarrow u \;\; a.e. $ in ...
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2answers
119 views

Checking whether an operator is self-adjoint. Problem with domain of an operator.

I want to check whether the position operator $A$, where $Af(x)=xf(x)$ , is self-adjoint. For this to be true it has to be Hermitian and also the domains of it and its adjoint must be equal. The ...
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1answer
35 views

Proving an operator $D: L^2[0,1]\rightarrow C'$, $Df(t)=\int^t_0 f(s) ds$ is unitary

Let $C'\subseteq C[0,1]$ be the space of all absolutely continuous function such that $f(0)=0$ and $f' \in L^2[0,1]$. Define an inner product on $C'$ as $\langle f,g \rangle = \int^1_0 f'(t)g'(t)dt$. ...
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2answers
1k views

Weak convergence in a Hilbert Space

What does it mean for a sequence $\{f_n\}_{n=1}^\infty\subseteq H$ to converge weakly? I know it means that it converges in the weak topology and I've read a few definitions of weak topology which all ...
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3answers
1k views

A Banach space that is not a Hilbert space

Can someone give me an example of a Banach space that is not a Hilbert space? I can't think of any because I don't know how to show one space that can not have inner product structure.
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1answer
182 views

right inverse of surjective linear map on hilbert space exists iff kernel is complement subspace

Suppose that $L : H \to H'$ is a surjective continuous linear transformation between Hilbert spaces. Show that there exists a continuous linear transformation $S : H' \to H$ such that $LS = I$. ...
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105 views

Riesz basis $\{|t|^\alpha e^{int}\}_n$

The class of Riesz bases is very large. It is extremely difficult to exhibit at least one bounded basis for a Hilbert space that is not equvalent to an orthonormal basis. We mention without proof the ...
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2answers
89 views

Examples of spectral decompositions

I would like examples of spectral decompositions and how they are obtained for normal compact operators and normal non-compact operators on an infinite dimensional hilbert space. I have googled it, ...
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1answer
54 views

Question about surjective continous operator being right invertible

I am reading a proof that a surjective continuous linear operator $T$ on a Hilbert space $H$ is right invertible. I have a question about the proof. The proof (up to the point where I have a question) ...
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171 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 ...
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1answer
172 views

The eigenvalues of a compact and self-adjoint operator on Hilbert space

Show that if $K$ is a compact self-adjoint operator on Hilbert space then it has either finitely many eigenvalues or a sequence of eigenvalues $\lambda_n\to 0$ as $n\to \infty$.
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1answer
67 views

Tychonoff vs. Hilbert

Let $(\mathscr H_n,\langle\cdot,\cdot\rangle_n)_{n\in\mathbb N}$ be a sequence of Hilbert spaces. Let $$\mathscr H\equiv\bigoplus_{n\in\mathbb N}\mathscr ...
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2answers
197 views

Questions about the proof of the Riesz representation theorem

Let $H$ be Hilbert space, $f:H \rightarrow \Bbb F$ linear and bounded map. I'm trying to prove that there exists only one $z_0 \in H$ such that: $ \forall_{x \in H} : f(x)=\langle x,z_0\rangle$ ...
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1answer
283 views

If every closed subspace of a Banach space has a closed orthogonal complement, then it is a Hilbert space.

My professor mentioned this fact in class. FACT: If every closed subspace of a Banach space has a closed orthogonal complement, then it is a Hilbert space. He mentioned that he had never seen the ...
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1answer
88 views

Prove that $A^2$ is an Hilbert Space.

We denote by $A^2$ the space of analytic functions on $B_1=\{z=x+iy\in \mathbb{C}, x,y\in \mathbb{R}||z|<1\}$, such that $$\left(\int\int_{B_1}|f(z)|^2 dx \, dy\right)^{1/2}<+\infty$$ In $A^2$, ...
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1answer
44 views

Inequality on Hilbert spaces in order to prove the nonexpansivity of a mapping.

I have an application $T\colon H\to H$ (where $H$ is a Hilbert space) such that $$(Tx-Ty,x-y)\leq \|x-y\|^2,\forall x,y\in H$$ where $(\cdot,\cdot)$ is the inner product of $H$ and $\|\cdot\|$ its ...
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1answer
51 views

Subspaces of finite dimensional Hilbert spaces

This might be a trivial question but please point out exactly where my reasoning is incorrect. Is every subspace of $\mathbb{R}^n$ closed since $\mathbb{R}^n$ with the dot product is a finite ...
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371 views

Homeomorphisms between infinite-dimensional Banach spaces and their spheres

As I know Cz. Bessaga has proved that an infinite-dimensional Banach space is homeomorphic to its unit sphere. Unfortunately I do not have his book but I want to know is this theorem true without ...
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1answer
90 views

On the isomorphism between bounded sesquilinear forms and bounded operators between two Hilbert spaces

Let $H$ and $K$ be two Hilbert spaces. Let $S(H,K)$ be the vector space of bounded sesquilinear forms $u:H\otimes \overline{K}\to\mathbb{C}$, and let $B(H,K)$ be bounded linear operators from $H$ to ...
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1answer
98 views

Confusion related to hilbert space

I was reading this article related to Hilbert spaces I didn't get why the first function space is not Hilbert space. I mean I can define the same norm $||f|| =\max_{a\leqslant x\leqslant b} ...
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1answer
134 views

Why is $L^2$ function Hilbert space not defined for Riemann Integral

The space of square Lebesgue integrable functions is said to be a Hilbert space. Why is if the integral is Riemann then this is not a Hilbert space? In other words, why not the space of Riemann square ...
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1answer
66 views

Surjective homomorphism example

What is an example of a surjective homomorphism $B(H)\to\mathbb C$, where $B(H)$ is the set of bounded linear operators on a Hilbert space $H$, and $\mathbb C$ is the complex numbers.
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1answer
82 views

Zero Operators on Complex Hilbert Space

This is a problem from Kreyszig's Introdcutory Functional Analysis with Applications. If for any $x$ in a complex Hilbert Space $<Tx, x> = 0$, show that $T\equiv 0$. Any clue?
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1answer
56 views

Gelfand transform explicity

Let $T$ be a bounded normal operator. Let $A$ be the algebra generated by $T$ and $T^*$. What is the explicit Gelfand transform $G:A\to C(\sigma(T))$? My book says the image of $T$ is the ...
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1answer
96 views

Expression for orthogonal projection onto Hilbert space (is related to Galerkin method)

Let $H=L^2(\Omega)$ and $V=H^1(\Omega)$. Suppose that $\{v_j\}$ is a basis for $H$ and $V$ (not necessarily orthogonal). Let $V_m = \text{span}(v_1, ..., v_m)$. Define a projection operator $P_m:H ...
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99 views

Hahn-Banach separation theorem for Hilbert spaces

What is the strongest form of the Hahn-Banach separation theorem for Hilbert spaces? Could you please provide a reference?
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654 views

Does this statement about Hilbert spaces make any sense?

I have found this tweet about git and don't know what to make of it. git gets easier once you get the basic idea that branches are homeomorphic endofunctors ...
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171 views

Difference between unconditional and absolute convergence in Banach spaces

One can show that in any finite-dimensional normed vector space absolute convergence is equivalent to unconditional convergence. It's not hard to show that if we have an orthonormal sequence in ...
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1answer
52 views

Direct product of a family of Hilbert spaces

Let $\{H_i\}_{i\in I}$ be a family of Hilbert spaces, defined $$H = \{f\in \Pi_{i\in I} H_i, \sum_{i\in I}|f(i)|^2<\infty\}$$ and inner product $\langle f,g\rangle : = \sum_{i\in I} \langle ...
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106 views

Reproducing kernel Hilbert sapce

I encountered the following claim (verbatim): Theorem Let $V$ be a subspace of $L^2(\mathbb{R})$ and $\{e_n\}$ be a orthonormal basis of $V$. The $V$ is a reproducing kernel Hilbert space with kernel ...
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1answer
63 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 ...
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1answer
39 views

Gelfand Transform in a specific case

What is the gelfand transform of an operator in the algebra generated by a bounded normal operator and it's adjoint? Thanks
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1answer
82 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, ...
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2answers
120 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 ...
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0answers
56 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. $S_\mu$ ...
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1answer
72 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 ...
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1answer
81 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 ...
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1answer
61 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 ...
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51 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 ...
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2answers
56 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 ...
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2answers
44 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 $\|T\|=\|T^*\|$?
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2answers
58 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 ...
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1answer
61 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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540 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 ...
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1answer
86 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 ...