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

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47 views

distance between a convex set and a point

Let's look at the following famous theorem: Let $\mathcal H$ be a Hilbert space and let $C< \mathcal H$ be a (proper) closed CONVEX set. If $x_0\in\mathcal H\setminus C$ and $\eta:=d(x_0, ...
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62 views

Closed linear subset of a Hilbert space

If $H$ is a Hilbert space, and if $$(a,b)_H=0$$ for every $b \in B \subset H$, where $B$ is a closed linear subset of $H$, does it follow that $a=0$, the zero element of $H$?
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34 views

$\langle f, v \rangle_{L^2(0,T;H'), L^2(0,T;H)}=0$ for all $v$ implies $f = 0$?

Suppose that for some $f \in L^2(0,T;H')$, $$\langle f, v \rangle_{L^2(0,T;H'), L^2(0,T;H)}=0$$ for all $v \in L^2(0,T;H).$ How do I show that this implies $f = 0$? $H$ is Hilbert.
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1answer
38 views

Basis for $L^2(0,T;H)$

Given a basis $b_i$ for the separable Hilbert space $H$, what is the basis for $L^2(0,T;H)$? Could it be $\{a_jb_i : j, i \in \mathbb{N}\}$ where $a_j$ is the basis for $L^2(0,T)$?
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61 views

Differentiating an infinite series in Hilbert space

Suppose $H$ is separable Hilbert space and $w_j$ is a basis. Suppose we have $h=\sum a_j(t)w_j$ an infinite sum where the coefficients are functions of $t$. The sum makes sense in the sense that the ...
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2answers
120 views

Characterisation of norm convergence

Let $X$ be a Hilbert space and $(x_n)\in X^{\mathbb N}$ be a sequence. Then the following statements are equivalent (with $x \in X$): We have $x_n \to x$, i.e. $\| x_n -x \| \to 0$ and we have $x_n ...
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79 views

On differential geometry in Hilbert spaces

Suppose that $H$ is a Hilbert space and $M\subset H$ is a closed subset with non-empty interior and smooth boundary, whatever smooth boundary could mean. I wonder if the normal vector is onto on the ...
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1answer
102 views

The definition of addition on the tensor product of Hilbert spaces

Let $H_1$ and $H_2$ be finite-dimensional Hilbert spaces with inner products $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$ respectively. Construct the tensor product of $H_1$ and ...
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3answers
589 views

What is a Hilbert space?

I've just seen a question about Hilbert Subspaces. This made me wonder what a Hilbert space is. Can anyone explain in layman's terms?
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171 views

Hilbert subspace

Let be $H$ Hilbert space and $M\subset H$. $M=M^{\perp\perp}$ if and only if $M$ subspace of $H$. Does anyone know to prove this?
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55 views

$\bigcup_{n}V_n$ is dense in $V$ implies $\bigcup_{n}L^2(0,T;V_n)$ is dense in $L^2(0,T;V)$?

Let $V$ be a separable Hilbert space with basis $w_j$ and let $V_n$ denote the linear span of $w_j$ for $j=1,...,n$. Clearly $V_n$ are Hilbert spaces and $V_n \subset V_{n+1}$ for all $n$. We have ...
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1answer
39 views

$\bigcup_{n}V_n$ is dense in $V$ (Hilbert spaces)

I read: $\bigcup_{n}V_n$ is dense in $V$ (Hilbert spaces) Does this mean: for every $v \in V$, there is a sequence $\{v_n\}$ with $v_n \in V_n$ for each $n$ such that $|v_n - v|_V \to 0$? I ...
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2answers
55 views

Question about finding minimum-Hilbert spaces

How to find $$\min_{a,b,c\in\mathbb{C}}{\int_0^{\infty}} |a+bx+cx^2+x^3|^2 e^{-x} dx = ?$$ Thanks in advance.
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1answer
32 views

Finding distance in Hilbert space

How to calculate $d(e_1,L)$, where $e_1=(1,0,0,\ldots)$ and $L=\left\{x\in l^2\mid x=(\xi_j)_{j=1}^\infty,\sum_{j=1}^n\xi_j=0\right\}$. Thanks in advance.
3
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1answer
351 views

A strictly positive operator is invertible

Suppose that $H$ is an Hilbert space, and $T: H \to H$ is a self-adjoint strictly positive operator (i.e. $\langle Tx,x\rangle > 0$ for all $x \neq 0$). How do I show that this operator is ...
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1answer
218 views

Operator self-adjoint

I have this paragraph : "Let M be a Hilbert-Riemannian manifold. $f \in C^2(M,R), p \in K$ is called a nondegenerate critical point, if $d^2 f (p)$ has a bounded inverse. Since $A = d^2 f (p)$ is a ...
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75 views

Is this operator bounded?

Let $w_j$ be a basis ( not orthogonal) of the Hilbert space $H$. For $h = \sum^\infty a_iw_i$ define $P_n(h) = \sum_{i=1}^n a_iw_i$. Is this operator bounded in $H$ I don't think it is but I feel ...
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2answers
258 views

Projection operator for non-orthonormal basis

Let $V \subset H$ be Hilbert spaces. Let $\{v_j\}_{j=1}^\infty$ be a basis for $V$ and $H$. Define $V_N$ to be the span of $\{v_j\}_{j=1}^N$. We can define a projection operator $P:H \to V_N$ by ...
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1answer
77 views

Does a continuous embedding preserve gaps between subspaces?

I have a separable, reflexive Banach space $(V,\|\cdot\|)$ that is continuously and densely embedded in a Hilbert space $(H,|\cdot|)$. This means, there is a bounded linear injection map $j\colon V ...
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1answer
78 views

How to prove that the set $e^{inx}$ is closed with respect to this measure?

Why is the set $\{e^{inx}\}$ closed in $L^2(d\nu)$, where $d\nu(x)=(1+|h(x)|)dx+|ds(x)|$? $d\nu$ is defined in this proof I am struggling to understand, where $\mu$ is a complex Borel measure on ...
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2answers
76 views

What is the correct definition of the cuspidal subspace of $L^2$?

I have a few (semi-)related questions regarding certain Hilbert space representations of locally compact groups that come up in the theory of automorphic forms. Let $G$ be a unimodular locally ...
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1answer
69 views

An operator on $H\times H$, with $H$ Hilbert

Let $(H, \langle \cdot,\cdot\rangle_H)$ a Hilbert complex space and consider $H\times H$ with the inner product $$\langle (u,v),(z,w)\rangle_{H\times H}\ =\ \langle u,z\rangle_H + \langle ...
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0answers
234 views

Direct sum of Hilbert spaces

Let $H$ be a separable Hilbert space with an orthonormal basis $\{e_n\}_{n =0}^{\infty}$. Consider a direct sum, $H \oplus H$. What is the orthonormal basis of $H \oplus H$ ? Is it $(e_n, e_m)_{n,m ...
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1answer
211 views

Proving Inner Product Space

Let $E=C^1 [a,b]$ be the space of all continuously differentiable functions. For $f,g \in E$ define $$ \langle f,g \rangle \ = \ \int_a^b f'(x) \ g'(x) \ dx$$ Is $\langle f,g \rangle$ an inner product ...
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0answers
179 views

expansiveness imply “relaxed monotonicity”?

Let $(H, \langle \cdot, \cdot\rangle)$ be a real Hilbert space and let $T:H\rightarrow H$ be a map. If there exists a constant $h>0$ such that $$\|Tx-Ty\|\geq h\|x-y\|, \quad \forall x,y\in H,$$ ...
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1answer
72 views

Weak convergence in $L^2$ and CDF

Assume that for sequence $X_n \in L^2(\Omega,F,P)$ which converges in distribution to CDF $F_X$ ($F_n(t)\rightarrow F_X(t)$ for every point of continuity of $F_X$), we have also that $X_n$ converges ...
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1answer
68 views

Inner product on a von Neumann algebra

Let $M$ be a $\sigma$-finite von Neumann algebra (one which admits a faithful normal state) acting on a Hilbert space $H$. Denote its faithful normal state by $\omega$. We can define an inner product ...
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168 views

Hilbert spaces - equivalent norm

Let $H$ be a Hilbert space with a norm $\| \cdot \|_1$. Let $\| \cdot \|_2$ be another norm on $H$ which is equivalent with $\| \cdot \|_1$. It is easy to see that $(H, \| \cdot \|_2)$ is a Banach ...
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1answer
40 views

A question about a limit in a Hilbert space

Suppose $H$ is a Hilbert space, $v_{n},z \in H$ and suppose that $$ \lim\langle x, v_{n}\rangle = \langle x, z \rangle$$ for all $x$ in some dense subset of $H$. Then can I say that the sequence ...
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57 views

Equivalent inner product on Hilbert space

Let $(H, (\cdot,\cdot)_1)$ be a Hilbert space. Suppose also that $(\cdot,\cdot)_2$ is an inner product on $H$ which is norm-equivalent with $(\cdot,\cdot)$. Is it possible to write the second inner ...
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2answers
148 views

Representing the tensor product of two algebras as bounded operators on a Hilbert space.

Hi Math StackExchange, Let $A$ be a commutative, infinite dimensional, unital, *-algebra represented by bounded operators on a Hilbert space $H_A$. Next let $B$ be a finite non-commutative *-algebra ...
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2answers
194 views

a trace class operator problem

Could someone help me with this Prove that If $A$ and $B$ are positive trace class operators on a Hilbert space, then so is $A^zB^{(1-z)}$ for a complex number $z$ such that $0 <Re(z)< 1$. An ...
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2answers
194 views

Proof of the Riesz Representation Theorem

Theorem: Let $F$ be a continuous linear functional on the Hilbert space $H$, then $\exists !$ (exists one and only one) $y \in H$ such that $F(x) = (x,y)$ for $x\in H$. Proof: Uniqueness: ...
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1answer
60 views

Surjective function on product space

I know that, if $U$ and $V$ are closed subspaces of a Hilbert $(X,\langle\cdot,\cdot\rangle)$, then these statements are equivalents: $$i)\ U^{\perp}\subseteq U+V\quad\quad\quad ii)\ X = ...
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1answer
187 views

Find best approximation in Hilbert space $H = L_2[0; 1]$ for element $a(t)$

I try to solve this problem all day, but can't reach any progress in it. Can you give me some hints? Find best approximation in Hilbert space $H = L_2[0; 1]$ for element $a(t) = \frac{1}{\sqrt[4]t}$ ...
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1answer
224 views

Criteria of compactness of an operator

Suppose $K$ is a linear operator in a separable Hilbert space $H$ such that for any Hilbert basis $\{e_i\}$ of $H$ we have $\lim_{i,j \to \infty} (Ke_i,e_j) = 0$. Is it true that $K$ is compact? ...
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33 views

Partial completion of subspaces of Hilbert spaces

Assume $H$ is a Hilbert space and $H_1\subset H$ and $H_2 \subset H$ are (closed) subspaces with $H_1 \cap H_2 = \{0\}$. Is there an $H_3 \subset H$, such that $H = H_1 \oplus (H_2 \oplus H_3)$ ? ...
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1answer
133 views

Hilbert space proof

$X$ is a separable Hilbert space and $ A\in L(X,X)$ and compact. I need to prove that $A$ is approximately of finite dimension.
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37 views

a question in projection

Let $V=L^2(\Omega)$, and $$k=\{v \in V ~s.t ~||v||_{L^2(\Omega)}\leq 1 \}$$ I need to find projection for any $u \in V$ on $k$. Please help me.I do not have any idea about this problem. I have many ...
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1answer
720 views

Show that Y is a closed subspace of l2

This might be a straight forward problem but I wouldn't ask if I knew how to continue. Apologies in advance, I am not sure how to use the mathematical formatting. We are currently busy with inner ...
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1answer
888 views

Bounded sequence in Hilbert space contains weak convergent subsequence

In Hilbert space $H$, $\{x_n\}$ is a bounded sequence then it has a weak convergent subsequence. Is there any short proof? Thanks a lot.
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1answer
120 views

Why can I choose elements of $X$ (and $h \in L^2(0,T)$) in this way? (Dual spaces, norms, Bochner spaces)

This is from the book Vector Measures by Diestel and Uhl, page 98: Let $X$ be a Banach space. Let $\epsilon > 0$ and suppose first that $g = \sum_{i=1}^\infty x_i^* \chi_{E_i}$ where $x_i^* \in ...
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1answer
75 views

Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary?

Is $H^1(M) \subset L^2(M) \subset H^{-1}(M)$ a Hilbert triple for $M$ a manifold with boundary? What smoothness is required of the boundary? I would be grateful for some references to this.
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90 views

Concerning unbounded linear operators on a Hilbert space

Let $H$ be some Hilbert space and let $B:H\rightarrow H$ be a bounded linear operator and $T:H\rightarrow H$ an unbounded linear operator. Furthermore we assume that $T$ is closed ,i.e. it's graph in ...
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1answer
100 views

Closest point property of subset of Hilbert space - what are the conditions for infimum to be finite?

I am proving the closest point property of a subset of a Hilbert space $H$: given $h\in H$ and a closed, nonempty and convex subset $M\subset H$, consider $$d=\inf_{m\in M} \|m-h\|$$ I am trying to ...
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2answers
286 views

What is my operator norm (cannot get good enough bounds).

Given a space of square integrable functions $x(t)$ over the interval $[0;1]$ one can introduce a norm $$\|x(t)\|= \sqrt{\int_0^1 (x(t))^2 \, dt};$$ Then what is a norm of the transformation below ...
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1answer
148 views

Example for a sequence of operators converging pointwise, but not with respect to the operator norm

I am trying to understand the following example. Define $$T_n: l^2 \rightarrow l^2$$ $$T_n(x)=(0, ..., 0, x_{n+1}, ...).$$ It's rather clear that $T_n(x)$ converges for $0$ for every $x \in l^2$. ...
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2answers
179 views

Hilbert space with all subspaces closed

Does there exist an infinite-dimensional Hilbert space with all subspaces closed?
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1answer
222 views

Weak convergence in Hilbert space L2 implies convergence in distribution?

Does weak convergence in $L^2$ (for $X_n, X \in L^2$ we say that $X_n$ converges weakly to $X$ ($X_n \rightarrow^w X$) if for every $Y\in L^2$ we have $\mathbb{E}X_nY \rightarrow \mathbb{E}XY$) ...
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1answer
145 views

Can a Accumulation Point be an Eigenvalue?

I have a discrete (separable) infinite dimensional Hilbert Space with a compact operator defined on it. So 0 is an accumulation point (some theorem says so). Can 0 also be an eigenvalue? And how would ...