Tagged Questions

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

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The meaning of an orthogonal basis?

I am reading up on Hilbert spaces and am a bit confused about the properties of an orthogonal basis. Would I be correct in saying that we can define an orthogonal basis as: Every element in the ...
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Vectors in Normed Space Must Have Finite Length?

I have assumed this to be the case, and consequently this is why one looks at convergent sequences of vectors in normed, Banach, and Hilbert spaces. But, I've never seen this listed explicitly as an ...
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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 ...