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

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Question about the Image of a compact transformation of a Hilbert space

$T$ is a compact operator on a Hilbert space. Show that $\operatorname{im}(T)$ does not contain a closed infinite dimensional subspace. Here is my attempt at the problem: Suppose that ...
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Direct sum of unitary operators is unitary

Is it true that if I have a Hilbert space $X$ that can be written $X = A \oplus B$ and unitary operators $T_1: A \to A$, $T_2: B \to B$ then the operator $T: A \oplus B \to A \oplus B$ given by ...
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Follow up on Hilbert spaces: chicken egg problem with projection and is the kernel always closed?

I am having a chicken egg problem with projections in Hilbert spaces. I was trying to show that if a Hilbert space can be written as $H = U \oplus U^\bot$ where $U$ is any subspace then $U$ is closed. ...
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38 views

On writing Hilbert spaces as sum of orthogonal complement

Is there a theorem that says: If $H$ is a Hilbert space and $U$ is any subspace then $$ H = U \oplus U^\bot$$ if and only if $U$ is closed? My conjecture is yes. I can easily prove that if ...
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Orthogonal and algebraic compement inclusion

Let $H$ be a Hilbert space. Let $U$ be a subspace and let $E$ be any complement such that $$ H = U \oplus E$$ I am wondering if it can be said that $U^\bot$ is contained in $E$. If not can it ...
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On Cauchy sequences, infimum and the proof of Hilbert space projection theorem

Let $H$ be a Hilbert space and $C$ a convex subset. The Hilbert space projection theorem states that there exists a unique $c_0\in C$ with $\|c_0\| = \inf_{c \in C}\|c\|$. I'm confused about the ...
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25 views

Tensor Product: Boundedness

This thread is just a note. Given Hilbert spaces. Then boundedness will be inherited: $$A,B\text{ bounded}\implies A\otimes B\text{ bounded}$$ Especially, the bounds multiply: $$\|A\otimes ...
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Tensor Product: Denseness

This is thread is just a note. Given Hilbert spaces. Then denseness will be inherited on tensor products: $$\mathcal{D},\mathcal{E}\text{ dense}\implies\mathcal{D}\otimes\mathcal{E}\text{ dense}$$ ...
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Tensor Product: Closability

This was a real question of mine. Given Hilbert spaces. Then closability will be inherited on tensor products: $$A,B\text{ closable}\implies A\otimes B\text{ closable}$$ For simple tensors this is ...
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Tensor Product: Identification

This is meant as note. Given a measure space and a Hilbert space. Then there's an identification: $$\mathcal{L}^2(\mu)\hat{\otimes}\mathcal{H}\cong\mathcal{L}^2_\mathcal{H}(\mu):\quad ...
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Tensor Product: ONB

This thread is just a note. Given Hilbert spaces. Consider their hilbertian tensor product: ...
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If the scalar product are equal then the operators are equal.

I want to show the following: Let H be a $\mathbb C$ -hilbert space and $S,T\in L(X)$ If $\langle Sx,x \rangle = \langle Tx,x \rangle$ for all $x\in H$, then $S=T$ Any hints for me?
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Showing $(M^\perp)^\perp=\overline{M}$

I have a question about a step in proving $(M^\perp)^\perp=\overline{M}$ where $M$ is a linear subspace of a normed vector space $E$. And $M^\perp=\{f\in E^*|\langle f,x\rangle =0\}$ This is the ...
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38 views

Invertible operators on a separable Hilbert space

Using polar decomposition or Kuiper's theorem one can show that the set of invertible operators on a separable Hilbert space $H$ is a connected subset of ${\mathcal B}(H)$. But does anyone know an ...
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32 views

Bergman space $L_a^2(\mathbb C)$

I claim that the Bergman space $L_a^2(\mathbb C)$ is the zero space. Is this true? If it is, how can I prove that every non-constant entire function is not in $L^2$?
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Symmetries on Hilbert spaces

Let $\mathfrak{H}$ be a Hilbert space and let $\mathcal{E}(\mathfrak{H})$ be the set of all operators $T\in B(\mathfrak{H})$ such that $0\leq T\leq 1$ (these operators are also called effects on ...
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Fourier multiplier is the only translation invariant bounded linear operator on $L^2[-\pi, \pi]$

This a question from Stein-Shakarchi Real Analysis. Let $\mathcal{H}= L^2[ -\pi, \pi]$. And define the $\textbf{Fourier Multiplier}$ by, $$Tf(x) \sim \sum_{-\infty}^{\infty} \lambda_n a_n e^{inx}$$ ...
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Orthogonal projectors on non-orthogonal subspaces

It is a well known fact that if(f) $V,W$ are orthogonal subspaces of a Hilbert space $H$, then their orthogonal projectors satisfy: $$ P_{VW} = P_V + P_W, $$ where $P_{VW}$ is the projector on $V+W$. ...
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Strong resolvent convergence and spectral measures

Suppose $\{A_n\}$ is a sequence of self-adjoint operators in a Hilbert space $\mathcal H$, and $A$ is a self-adjoint operator, with $A_n \to A$ in the strong resolvent sense. Denoting by $E_n$ and $E$ ...
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A vector space with norm metric induced by inner product

What is the meaning of " A vector space with norm metric induced by inner product ? " Can norm metric induced by opertion(s) other than inner product ?
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Is a bounded operator with finite trace trace class?

Let $\mathcal{H}$ be a seperable Hilbert space, $A\in\mathcal{B}(\mathcal{H})$ a bounded linear Operator and assume we have an orthonormal basis $(x_n)_{n=1}^\infty$. If $A$ is trace-class, then ...
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References on projectors

What are good books or articles about linear projectors in Hilbert spaces? I am mostly interested in the finite dimensional case (but anything is welcome). All about idempotents, orthogonal and ...
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Is $X= \{ u \in H^1(\Omega \times I) \mid \int_\Omega u(x,y)\;\mathrm{d}x = 0 \text{ for a.a. $y \in I$}\}$ a Hilbert space?

Let $\Omega$ be a bounded domain and let $I$ be an unbounded interval. Let $$X= \{ u \in H^1(\Omega \times I) \mid \int_\Omega u(x,y)\;\mathrm{d}x = 0 \text{ for a.a. $y \in I$}\}$$ Is this ...
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Approximation Property: Decomposition

This is a real question of me. Given a Banach space $E$. Consider a finite rank operator $F\in\mathcal{F}(X,E)$. Introduce a basis on the finite dimensional range: ...
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How to prove this sequence is in $l^2$? [duplicate]

I ran into such a problem in some exercise book on hilbert space. Suppose we have a sequence $ \{a_n \}_1^\infty$. Now, for any sequence $\{ b_n \}_1^\infty $ in $l^2$, we have $$ ...
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Short proof of sequential Banach Alaoglu for Hilbert spaces

Do you know of a short proof of the fact that bounded sequences in Hilbert spaces admit weakly converging subsequences? If the space is separable, then the common sequential-version proof is what I ...
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Embedding $L^2[0,1]$ into any Hilbert space?

Is it true that every Hilbert space has a closed subspace isometrically isomorphic to $L^2[0,1]$? Can someone sketch a proof of this, or at least point me in the right direction to understanding it? ...
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40 views

Inverse operator of $I-A$

Let $H$ be an Hilbert space, $A:H\to H$ be a bounded linear operator such that $$ \|A^{n_0}\|< 1\qquad\text{for some}\quad\; n_0\in\mathbb{N}. $$ I have to show that $I-A$ is invertible. My idea ...
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23 views

Partial Isometries: Subspaces

Note: This thread is not to gain reputation!!! Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider a bounded operator: $$W:\mathcal{H}\to\mathcal{K}:\quad\|W\|<\infty$$ Then a partial ...
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Adjoints of operators between different Hilbert spaces.

When we have an operator $$ T ~\colon \mathscr{H} \longrightarrow \mathscr{H} $$ from a Hilbert space to itself, we can use the Riesz representation theorem to prove the existence of the adjoint ...
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Open set in Hilbert Cube.

Any open set in the Hilbert Cube is the union of open subsets of the form $$U_1 \times ... \times U_n \times X_{n+1} \times .... \times X_{n+k} \times...$$ where $X_k := [0, \frac{1}{k}]$ for $k \in ...
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Action of projections

Suppose we have a projection $p$ on a Hilbert space $\mathfrak{H}$. Is the following true: There exists an set $V\subset\mathfrak{H}$ such that $p(x)=x$ if $x\in V$ and zero else? I asked because I ...
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Strong Topology and Strong Operator Topology on Hilbert Space

Suppose $H$ is a Hilbert space (much of this still works if it's just a Banach space), $x\in H$, and $(x_n)$ a sequence in $H$. Does $x_n\to x$ strongly in H iff $x_n\to x$ as operators in the strong ...
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Orthogonality of projections on a Hilbert space

Assume that $p$ and $q$ are (orthogonal) projections on Hilbert space $\mathcal{H}$. I want to prove: $pq=0$ iff $p+q\leq1$ I had the following in mind: Assume $pq=0$. Then $qp=0$, hence $p+q$ is a ...
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Is the composition function again in $L^2[a,b]$ [closed]

Let $f \in L^2[a,b]$. 1- In what condition(s) on a function $g:[a,b]\rightarrow [a,b]$ we can get $$f \circ g \in L^2[a,b]?$$ 2- In what condition(s) on $g:[a,b]\rightarrow [a,b]$, the operator ...
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51 views

Difference between total orthonormal set and basis

I'm learning about Hilbert spaces and related things from the book "Introductory functional analysis with applications". Now I just read the following sentence, which I don't quite understand: "A ...
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416 views

Infinite Dimensional Hilbert Space

Let $H$ be a Hilbert space with a countable basis $B$. Does it mean that any vector $x\in H$ can be expressed as a finite linear combination of elements from $x$, or as an infinite linear combination? ...
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Prove that space of Laurent series is not Hilbert

Let $z_0 \in \mathbb{C}, s>0,T(z_0,s):=\{z\in\mathbb{C}:|z-z_0|=s\}$ and let $V=V(z_0,s)$ be a vector space over field $\mathbb{C}$ of all Laurent series that are uniformly and absolutely ...
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What is the norm on the completion of a Hilbert space?

Let $X$ be a Hilbert space with a norm $|u|_X = |u|_{X_1} + |Gu|_{X_1}$, where $G:X \to X_1$ is linear and continuous, $X_1$ is a Hilbert space. Define $$|u|_Y = |Gu|_{X_1} + |Tu|_{Z}\quad\text{for ...
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How to prove that any Hamel basis of an infinite-dimensional complete and separable real inner-product space is uncountable?

How to prove that any Hamel basis of an infinite-dimensional complete and separable (having a countable dense set ) real inner-product space is uncountable ? Do I have to use Baire-category theorem ? ...
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Does every closed, densely operator in a Banach space have an closed, densely defined extension on a Hilbert space?

Assume that $H_1,H_2$ are separable Hilbert spaces, $B$ is a separable Banach space and $H_1\subset B\subset H_2$. Assume further that the inclusion mappings are continuous and have dense images. ...
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Møller Operators: Summary

Disclaimer This thread is meant as summary. For more informations see: SE blog: Answer own Question MSE meta: Answer own Question (The second especially reveals the opinion of the community!) ...
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Spectral Measures: Equivalence

This thread is a record. Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}\to\mathcal{H}:\quad N^*N=NN^*$$ and its spectral measure: ...
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Spectral Measures: Restriction

This thread is just a note. Given a Hilbert space $\mathcal{H}$. Consider a normal operator: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ Denote its spectral measure by: ...
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Spectral Measures: Uniqueness

Given a Hilbert space $\mathcal{H}$. Consider spectral measures: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ $$E':\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ How to prove ...
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infinite sum of normal r.v. is still a normal r.v. when given $\sum \limits_{i=1}^\infty a_i^{2}$ is finite

If $X_1, X_2, ...$ are i.i.d.standard normal random variables and for real constants $a_1, a_2, ...$, given $\sum \limits_{i=1}^\infty a_i^{2} $ is finite, then $Y_n =\sum\limits_{i=1}^n a_iX_i$ ...
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Hilbert space projection theorem: how to finish my proof?

The Hilbert space projetion theorem is the following theorem: Let $H$ be a Hilbert space and $C$ any closed convex subset. Then for $h \in H$ there exists a unique $c_0 \in C$ such that $\|h-c_0\| = ...
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Dynamics: Continuity

Disclaimer: This is a record of results. Given a C*-algebra $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$. Consider a Hamiltonian dynamics: ...
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Discrete J-method of interpolation (about understanding theorem statement)

The discrete $J$ method is, given Banach spaces $A_0$ and $A_1$: The interpolationn space $[A_0, A_1]_\theta$ is defined by: $a \in [A_0, A_1]_\theta$ if and only if $a$ can be written as ...
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$f_n\to f$ in $L^2$ and $fg\in L^2(\Omega)\implies f_n\,g\in L^2?$

Let $f,g\in L^2(\Omega),\,$ $f_n\in L^p\,\,\forall 1\leq p<\infty$ such that $f_n\to f$ in $L^2$ and $fg\in L^2(\Omega)$. I was trying to understand if we can derive that $f_n\,g\in L^2?$ My first ...