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

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Is every finite dimensional linear space a banach space [on hold]

Is every finite dimensional linear space a Banach Space? Is every finite dimensional linear space a Hilbert Space?
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1answer
13 views

Compact Operators: Approximation Property

Note: This thread is not to gain reputation!! Given a Hilbert space. It has the approximation property!
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1answer
30 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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1answer
14 views

Partial Isometries: Subspaces

Note: This thread is not to gain reputation!!! Given an operator algebra. Then a partial isometry satisfies both: ...
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1answer
16 views

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 ...
3
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1answer
16 views

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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1answer
17 views

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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1answer
13 views

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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1answer
29 views

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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2answers
37 views

Is the composition function again in $L^2[a,b]$ [on hold]

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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1answer
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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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2answers
375 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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1answer
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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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0answers
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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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2answers
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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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1answer
14 views

Spectral Measures: Reducing Subspaces

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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0answers
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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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1answer
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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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2answers
42 views

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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1answer
27 views

Spectral Measures: Restriction

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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0answers
19 views

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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2answers
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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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1answer
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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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0answers
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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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1answer
39 views

$f_n\to f$ 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 ...
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2answers
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How can I show $U^{\bot \bot}\subseteq \overline{U}$?

Let $H$ be a Hilbert space and $U$ a subspace. Let $U^{\bot}$ denote its orthogonal complement. I had no trouble showing $\overline{U}\subseteq U^{\bot\bot}$. But now I'm stuck for $\supseteq$. ...
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1answer
21 views

Closure of the image is equal to image of $u^\ast u$?

Let $u \in B(H,H')$ where $H,H'$ are Hilbert spaces and let $u^\ast$ denote its adjoint. How can I see that $\overline{u^\ast(H')} = u^\ast u (H)$?
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What do mathematicians mean by “equipped”

I am a mathematical illiterate so I do not know what people mean when they say equipped. For example, I say that Hilbert space is a vector space equipped with a inner product. What does that ...
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1answer
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Why is $\langle x-P(x),m\rangle=0$?

Let $H$ be a Hilbert space, and let $M\le H$ be a subspace of it. Let $P:H\rightarrow M$ be the orthogonal projection $H$ onto $M$. We'll take $x\in H$, and $m \in M$. By the definition I know ...
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1answer
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Significance of closedness of a subspace when writing a Hilbert space as a direct sum

I read that if $U$ is a closed subspace of a Hilbert space $H$ then we can write $H$ as $H = U \oplus U^\bot$ (the direct sum). What is not clear to me is why $U$ is required to be closed. I thought ...
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1answer
66 views

Questions about the Fourier transform as a unitary transform

As far as I know, the Fourier transform is a (linear) unitary transform: $T: \textbf{L}^2(-\infty, +\infty) \rightarrow \textbf{L}^2(-\infty, +\infty)$ where the basis functions {$e^{i \omega x} | ...
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2answers
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Prove $\big|\langle x,y \rangle\big| \space ≤ \space \lambda \cdot \|x\|^2+\frac{1}{4\lambda} \cdot \|y\|^2$ in an inner product space

I want to prove that if I have an inner product space with $\lambda>0,$ then $$\big|\langle x,y \rangle\big| \space ≤ \space \lambda \cdot \|x\|^2+\frac{1}{4\lambda} \cdot \|y\|^2$$ Where should I ...
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0answers
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Sum of closed subspaces of a Hilbert space is closed

Let $M, N ⊂ H$ ($H$ Hilbert), be two closed linear subspaces. Assume that $\langle u, v\rangle = 0$ $∀u ∈ M$, $∀v ∈ N$. Prove that $M + N$ is closed. Take a sequence $(g_n)\in M+N$ such that $g_n\to ...
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1answer
36 views

Orthogonal projection on subspace

Let $\Omega$ be a measure space and let $h : \Omega → [0, +∞)$ be a measurable function. Let$$K = \{u ∈ L^2(\Omega);\ |u(x)| ≤ h(x)\ a.e. on\ \Omega\}.$$ Check that K is a non-empty closed convex ...
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2answers
39 views

Hilbert space Inequality

Let $K ⊂H$ be a nonempty closed convex set, $H$ Hilbert. Let $f ∈H$ and let $u=P_Kf$. Prove that $$||v − u||^2 ≤ ||v − f ||^2 − ||u − f ||^2, ∀v ∈ K$$ I've tried to use the parallelogram identity and ...
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1answer
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Orthogonal complement of a subspace in a Hilbert space

In this question it is stated that if $W$ is a closed subspace of $V$, we may define $W^\perp$ to be the subspace of all vectors $v \in V$ such that $\langle v | w\rangle =0$ for all $w \in W$. ...
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0answers
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Spectral definition of (fractional) Laplacian, need help understanding text

Let $\varphi_k$ and $\lambda_k$ be the eigenfunctions and eigenvalues of the Dirichlet Laplacian $-\Delta$ on some bounded domain $\Omega$. We know $\varphi_k$ are smooth and form an orthogonal basis ...
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2answers
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Proving compactness of an operator

I'm having a hard time finding a solution for the following problem: Prove that the operator $ T \in \mathcal{L}(\ell_2) $ defined with the formula $$ T((x_1, x_2, \dots, )) = (0, x_1, x_2/2, x_3/3, ...
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1answer
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Strict Bessel inequality in $\mathcal{l}^2$

I'm asked to give an example of an $x\in \mathcal{l}^2$ s.t. $$\sum_{j=1}^{\infty}|<e_j,x>|^2<||x||^2$$ Where $(e_j)$ is some orthonormal sequence. However, I think this question is more ...
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1answer
27 views

General Fourier coefficients and smoothness

Suppose $f\in L^2([0,1],\lambda)$. Are there assumptions on the smoothness of $f$ which translate into the particular behavior of Frourier coefficients. Namely, I have arbitrary complete orthonormal ...
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1answer
20 views

Equivalent finite subspaces of a hilbert space

I have to prove the following statement: Let $H$ be a Hilbertspace and $M,N$ closed subspaces. Then the following holds: If $M \sim N $ and $N$ is finite, then $M$ is finite. I think it should say ...
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2answers
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Fock Space: KMS-State

Given the CAR-algebra with Hamiltonian dynamics: $$\tau^t[a^\#(\eta)]=a^\#(e^{itH}\eta)\quad(H:\mathcal{D}\to\mathcal{H})$$ (Caution that the Hamiltonian is usually unbounded.) Consider a KMS-state: ...
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1answer
34 views

Why does this completion of a Sobolev space contain constant functions? Please explain text.

Below, $\mathcal{C} = \Omega \times (0,\infty)$, $x$ refers to the variable in $\Omega$ and $y$ to the variable in $(0,\infty)$, and $\Omega$ is a bounded smooth domain. $tr_\Omega:H^1(C) \to ...
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2answers
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Fock Space: Formal Adjoints

Problem Given a pre-Hilbert space $\mathcal{H}$. Consider unbounded operators: $$S,T:\mathcal{H}\to\mathcal{H}$$ Suppose they're formal adjoints: $$\langle ...
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What is the idea behind interpolation spaces?

I am working through a text on Numerics for SPDEs and there the concept an interpolation (Hilbert-)space associated to an operator is used. To be specific: Definition. Let $H$ be an ...
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When $1 \le p \le \infty, p\ne 2$, $L^p$ space is not a Hilbert space

It suffices to show that when $1 \le p \le \infty, p\ne 2$, $L^p$ norm does not arise from an inner product.(there is a hint saying that we can use the parallelogram law) I can proof a special case ...
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0answers
34 views

Fourier transform and non-standard calculus

The Fourier transform is not as easily formalizable as the Fourier series. For example, one needs to introduce tempered distributions to define the Dirac delta-function. Also, it is impossible to view ...
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1answer
39 views

Prove that if ,$||f||^2 = A\sum_{j}|<f, \phi_j>|^2 $ then $f = \sum_{j}<f, \phi_j> \phi_j$

Let $\phi_k$ be some sequence of real functions in an infinite Hilbert space $H$ such that there exists $A \in \mathbb{R}$ such that for all $f \in H$ ,$||f||^2 = A\sum_{j}|<f, \phi_j>|^2 $ ...
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1answer
22 views

For a given Hilbert space find a tight frame with bound A

For a given Hilbert space and $A>0$ find a tight frame with bound A. I know that an ortho-basis is a tight frame with $A=1$. Can I extend this to any $A>0$ by just scaling the ortho-basis?