# Tagged Questions

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

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### Fatou lemma and weak convergence in Hilbert

In a Hilbert space $H$ a sequence $(x_n)_{n\geq0}$ is said to converge weakly to $x$ if $\forall y\in H:\langle y,x_n\rangle\rightarrow\langle y,x\rangle$, the case in which we can easily deduce an ...
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### Distance preserving function on hilbert space

It is known that an isometry on B(H) is distance preserving .I am trying to show ,conversely , that if F=R,every distance preserving function f on H( Hilbert space) has the form f(x) = f(0) +Tx for ...
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### Conjugate linear isometry of Hilbert operators

Let $H$ and $R$ be Hilbert spaces and consider an operator $T$ in $B(H,R)$. I need to show that there is a unique operator $T^*$ in $B(R,H)$ satisfying $$(Tx│y)_R = (x│ T^* y)_H,$$ $x \in H$, $y \in R$...
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### Generalized polar decomposition

Let $x\in B(H)$. We say $(x,v,y)$ is a polar decomposition for $x$ if, $\bullet$ $y$ is positive. $\bullet$ $v$ is a partial isometry with $x=vy$. $\bullet$ Ker$(x)$=Ker$(y)$=Ker($v$) The polar ...
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### Let $H$ be a Hilbert space, $V≤H$ be closed, $Q:H→V$ be the orthogonal projection, $(e_n)_{n∈ℕ}$ be an ONB of $H$. Is $(Qe_n)_{n∈ℕ}$ an ONB of $V$?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$ $U$ and $H$ be $\mathbb K$-Hilbert spaces $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $\iota:U\to H$ be an embedding and $V:=\iota(U)$ ...
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### Orthogonal sequences under A Hillbert space

I know that for two vectors $u,v\in H$ where $H$ is a Hilbert space the definition for orthogonality is $\langle u,v \rangle =0$. is thaat also corret for sequences? What is the definition for ...
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### Prove a sequence is bounded under a Hilbert space

Let $T:H\to H$ be defined by $Tx=\sum_{n=1}^\infty \lambda_n \langle x,\varphi_n \rangle \varphi_n$ where $\{\varphi_n\}_{n=1}^\infty$ is an orthogonal sequence and $\{\lambda_n\}_{n=1}^\infty$ is a ...
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### Show that $\ker(T)=\{\varphi _n\mid\lambda_n\neq 0\}^\perp$

Let $T:H \to H$ be defined as $Tx=\sum_{n=1}^{\infty} \lambda_n \langle x,\varphi _n \rangle \varphi _n$, given that $\{\varphi _n\}_{n=1}^\infty$ is an orthonormal sequence (not necessarily a basis) ...
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### A factorization for operators

Let $a$ be an arbitrary operator in $B(H)$ and $b$ be a positive operator in $B(H)$. Assume $a$ and $b$ have the same null space and there exists an operator $u\in B(H)$ with $a=ub$. Q) Can we ...
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### I don't understand how the adjoint operator is used in a book that I'm reading

I'm reading Stochastic Differential Equations in Infinite Dimensions and don't understand what the authors do in Chapter 2.3.1. Let me introduce the necessary objects: Let $K$ and $H$ be real ...
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### Operator with norm

I got the following problem to solve: Let $H$ Hilbert space and $T: H \to H$ a bounded positive operator, i.e. \begin{align*} \langle x, T x \rangle \geq 0 & & \text{for all } x \in H. \end{...
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### Bounded operators on inner products on Hilbert space

If we have a Hilbert space $H$ with inner product $( \cdot | \cdot)$, and let $( \cdot| \cdot)_1$ be another inner product on $H$ such that $(x | x)_1 \leq (x | x)$ for every $x \in H$. I was trying ...
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### Convergence of unitary products on a Hilbert space

First: I'm sorry for the basic question--I can move it to Math SE if necessary... Let $X$ be a Hilbert space and suppose $\{U_k\}_k$ is a sequence of unitary operators on $X$. Let $\|\cdot\|$ be the ...
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### Proof about orthogonality of columns of a matrix

Consider a matrix $A \in \mathbb{R}^{n \times n}$ and the canonical inner product in $\mathbb{R}^{n}$. Show that if the rows of A form an orthogonal set, the same happens with the columns. So what ...
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### Hausdorff-Quotient: Embedding

Problem Given a uniform space $\Omega$. (Exemplary Topological Vector Space!) Consider a dense subspace: $$\iota:\mathcal{D}\hookrightarrow\Omega:\quad\overline{\iota\mathcal{D}}=\Omega$$ Regard ...
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### Convexity of Hilbert cube [closed]

I am trying to show that the Hilbert cube $\{ x_n \in l^2(\mathbb{N}) \mid x_n \in [0, \frac{1}{n}] \ \forall n \in \mathbb{N} \}$ is convex and (norm)-compact.
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### Cardinality of a Hilbert space

I have seen the theorem about the cardinality of orthonormal basis of a Hilbert space. I wonder if we have a Hilbert space $H$ with an orthonormal basis having cardinality of the continuum, then what ...
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### Separable infinite-dimension Hilbert space and its subspaces

Suppose $H$ is any separable infinite-dimensional Hilbert space. Then $H$ has family of closed subspaces $\big\{ E_t :~ t \in [0,1]\big\}$ such that $E_s$ is a strict subspace of $E_t$ for all \$0 \leq ...