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

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1answer
129 views

Operator norm and spectrum

Let $L$ be a bounded linear operator on a Hilbert space. Without assuming finite dimensions, can we express the operator norm of $L$ in terms of the spectrum of the positive operator $L^{\dagger}L$? ...
5
votes
1answer
319 views

Weak Formulations and Lax Milgram:

I have a question on how to put a PDE into weak form, and more importantly, how to properly choose the space of test functions. I know that for an elliptic problem, we want to start with a problem ...
3
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2answers
105 views

Show the subspace $\textrm{span}\{e_n:n\in\mathbb N\}$ is not closed in $\ell^2$

How to show the subspace $\textrm{span}\{e_n:n\in\mathbb N\}$ where $e_n=(\delta_{nk})_{k\in\mathbb N}$ is not closed in $\ell^2$?
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3answers
42 views

Dimension of a space

I'm reading a book about Hilbert spaces, and in chapter 1 (which is supposed to be a revision of linear algebra), there's a problem I can't solve. I read the solution, which is in the book, and I ...
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2answers
613 views

Tensor product of operators

We know that if $T_1$ is a linear bounded operator on a Hilbert space $H_1$ and $T_2$ is a linear bounded operator on a Hilbert space $H_2$ there exists a unique linear bounded operator $T$ on $H_1 ...
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2answers
96 views

isomorphic properties

I need help with this proof: When $L$ is a isomorphic (bijection) linear mapping between two Banach spaces , in the case of both the spaces are Hilbert when using $L$, is it right to say that the ...
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1answer
176 views

Orthonormal basis in Hilbert spaces

I have a general question but I'm going got ask it in a very restrictive setup. It is known that an equivalent condition for a system $\left\{e^{i\lambda t}\right\} _{\lambda\in\Lambda}$ being an ONB ...
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1answer
89 views

Norm of element of Hilbert space

How to prove that in a Hilbert space $H$, $$\lVert h \rVert = \sup_{u \in H}\frac{|(h,u)|}{\lVert u \rVert}?$$ Showing that the RHS is $\leq$ the LHS is easy but not sure of the other part. This is ...
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2answers
453 views

compact and self adjoint square root of an operator

Let H a Hilbert space and $T:H\rightarrow H$ a linear bounded, self-adjoint, positive and compact operator. How can i prove that the square root of T, $\ T^{1/2}:H\rightarrow H$ is also compact and ...
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1answer
127 views

Infimum of a Hilbert space inner product

This is exercise 5.11 in Brezis's Functional Analysis, Sobolev Spaces, and PDEs. Let $H$ be a Hilbert space, and let $M \subset H$ be a nonzero closed linear subspace. Let $f \in H$, $f \notin ...
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1answer
263 views

Riesz representation theorem on dual space

We all know the Riesz representation theorem, so we can represent a bounded linear function on a Hilbert space $H$ with an inner product on $H$ and vice-versa. My question is, given an inner product ...
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1answer
133 views

Multiplication not continuous in $B(H)$ in the strong operator topology

I'm reading this answer by t.b., and I'm only interested in the case when $X$ is an infinite-dimensional Hilbert space. Regarding Question 2, in the first bullet point he claims the following: "Given ...
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2answers
122 views

Does $e_n(x)=\exp\left( \frac{i \pi n}{N}x \right)$ define an orthonormal basis of $L^2(-N,N)$?

We know that the Fourier system is complete, i.e. that $\lbrace e_n: ~ n \in \mathbb{N} \rbrace$ defined by \begin{equation} e_n(x)=\frac{1}{\sqrt{2 \pi}}\exp(inx), ~~~ n \in \mathbb{Z} \end{equation} ...
9
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1answer
167 views

Is the identity map $id: H^2(-\pi,\pi) \to L^2(-\pi,\pi)$ Hilbert-Schmidt?

Let $H_1, H_2$ be Hilbert spaces. A linear operator $A: H_1 \to H_2$ is Hilbert-Schmidt iff for some orthonormal basis $\lbrace e_n : ~ n \in \mathbb{N} \rbrace$ of $H_1$ the sum $\sum_{n \in ...
5
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1answer
64 views

Space of Jordan curves

The space of square-integrable functions $f:[0,1]\rightarrow\mathbb{R}$ is well conceivable: it's essentially an $\infty$-dimensional Euclidean space (the Hilbert space $L^2$) with well interpretable ...
4
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1answer
116 views

functional analytic interpretation of the (co)variation and the doob decompostion

I have a question concerning the covariation of two time-discrete stochastic processes. Let $(\mathcal{F}_i)_{i\in T}$ be a filtration. We call a time-discrete, real-valued, adapted process $X$ ...
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1answer
111 views

Projection on a Hilbert space

Can anyone help me with this proof. Let $M=\{e_1, e_2, \ldots\}$ be an orthonormal subset of a Hilbert space $H$ and $A=\overline{\textrm{span}(M)}$. Show that the orthogonal projection ...
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2answers
150 views

Is duality an exact functor on Banach spaces or Hilbert spaces?

Let $V,V',V''$ and $W$ be vector spaces over $k$. Then, it is known that $\operatorname{Hom}(\cdot,V)$ is a contravariant exact functor, i.e. for each exact sequence $0\to V'\to V\to V'' \to 0$, and ...
2
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1answer
48 views

Is the adjoint of a quasinormal operator quasinormal as well?

I am trying to make sense of the various properties of operators on Hilbert spaces that generalise the notion of normality. It is known that for a (bounded) operator $A$ there are the following ...
3
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1answer
102 views

Why define a function space in this fashion? (PDE and functional analysis)

Suppose that for $t \in [0,1]$, $X(t)$ is a Hilbert space of functions, eg. $X(t) = L^2(\Omega_t)$ where $\Omega_t$ is a bounded domain. Define a space $$H := \{\overline{v}:[0,1]\to \bigcup_{t ...
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1answer
81 views

How to show projection of $L^2$ function converges to that $L^2$ function

My teacher said that if $P_n f = \sum_{j=0}^n(f,w_j)w_j$, where $w_j$ is orthonormal basis of $L^2$, then $|P_n f- f|_{L^2} \to 0$ for $f \in L^2$. How do I prove this? I thought $$|P_nf - f| = ...
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1answer
92 views

Is an orthnormal basis of $L^2([0,1])$ also an orthonormal basis of $L^2((0,1))$?

My question is: If $\lbrace e_n \rbrace$ is an orthnormal basis of $L^2([0,1])$, is $\lbrace {e_n}_{|(0,1)} \rbrace$ an orthonormal basis of $L^2((0,1))$? As the points $\lbrace 1 \rbrace$ and ...
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2answers
131 views

Uncountable union of separable spaces is separable?

If $V(x)$ is a separable Hilbert space, is $\bigcup_{x \in X}V(x)\times\{x\}$ separable when $X$ is an uncountable set? How to make it separable if it's not? What assumptions do I need?
4
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1answer
59 views

Selfadjoint and continuous operator on a complex Hilbert space

Let $T\colon H\to H$ be a selfadjoint continuous operator on a complex Hilbert space. Show: $$ \lVert (T\pm i\mbox{Id})x\rVert^2=\lVert Tx\rVert^2+\lVert x\rVert^2~\forall~x\in H. $$ -- How can I ...
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2answers
414 views

Isomorphism of Banach space

If $T:H\to B$ is isomorphism of Banach spaces and $H$ is Hilbert, must $B$ necessarily be Hilbert?
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0answers
127 views

Find the adjoint operator

I would like to find the adjoint operator in the Hilbertspace $L^2(0,\infty)$ of the operator $$ (Ax)(t)=x(at), x\in L^2(0,\infty), a>0. $$ My calculation is the following; I use the ...
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2answers
566 views

Parallelogram law valid in banach spaces?

It is known that the parallelogram law $\|x-y\|^2+\|x+y\|^2 = 2(\|x\|^2 + \|y\|^2)$ holds in any space with an inner product (the norm being induced by this inner product). Is this formula valid in ...
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2answers
291 views

Equivalence of norms implies isomorphism between Hilbert spaces

If I have 2 Hilbert spaces with 2 norms, and a map between the Hilbert spaces, and I know that the norms are equivalent, does this mean that the spaces are isomorphic?
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0answers
43 views

Hilbert basis of $L^2([-1,1])$?

Could you please specify hilbert basis of $L^2([-1,1])$? How will be the representation of a function f $\in L^2([-1,1])$ by means of its Fourier series? My solution: $E_k=1/\sqrt2 e^{kit\pi}, k\in ...
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1answer
39 views

$V \subset H \subset V^*$, what's $\langle h, v \rangle_{V^*, V}$?

If $V \subset H \subset V^*$ is Hilbert triple, and $h \in H$ what's $\langle h, v \rangle_{V^*, V}$? I know we interpret it to be $(h,v)_H$. But is this correct: $$\langle h, v \rangle_{V^*, V} := ...
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1answer
94 views

Closed subset of a Hilbert space

$Y_0\subset Y $ is a closed Hilbert subspace of $Y$ with finite codimension and a subspace $ Y_1 $ satisfies $ Y_0 \subset Y_1 \subset Y $. Is $ Y_1 $ also closed?
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2answers
208 views

Trying to understand Hilbert Spaces…

I am trying to get a hold on Hilbert Spaces, but I am having difficulties combinging various definitions. I have looked it up on wikipedia and wolfram, there it states something like "A Hilbert ...
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0answers
116 views

Construct an orthonormal basis

Consider vector space of functions continuous on $I \subseteq \mathbb R$ and scalar product is defined as $({\bf f}, {\bf g}) \equiv \int_{I} \rho(x) f(x)g(x) dx$. Let the generating function $G(z,x)$ ...
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1answer
298 views

Estimate on the norm of a self-adjoint operator

EDIT: thks to Martin's comment I realize the previous version was wrong. Here is the correct version of what I need to show: I am trying to show that if $A$ is a self - adjoint operator in a Hilbert ...
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1answer
128 views

proof of RKHS for a particular kernel is unique

Suppose that I have a kernel $K$. Then show that the RKHS $H_1$ and $H_2$ of $K$ are the same. So I need to prove the above statement. To begin with, as an exercise, I proved the reverse statement ...
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1answer
135 views

Matrix Trace representation?

For a real, symmetric matrix $A$ and a real, rectangular matrix $X$, am looking for a matrix trace based representation of this simple linear algebraic expression $\sum_{i} A_{ii} ...
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2answers
81 views

Given $F:X \to X$ on $X$:Hilbert space satisfying some properties, prove that $F$ is surjective.

Given a map $F:X \to X$ where $X$ is a Hilbert space, $F$ satisfying $f(x):=x-F(x)$ is a compact map. $\lim_{\|x\|\to \infty} \frac{(F(x),x)}{\|x\|} = \infty$ I'm seeking to prove that $F$ is ...
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2answers
160 views

elliptic pdes and associated bilinear forms for Lax-Milgram

I have a simple question on elliptic pdes, actually I can not understand clearly from definitions. Thats why I want to try think on an example. Let us have an elliptic pde $$-A \Delta ...
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1answer
46 views

Kernel inclusion implies factorization

I have a question whether a certain fact is true for arbitrary operators on a Hilbert space. Namely, consider Hilbert spaces $H,K$, an operator $A\in B(H)$ and another $B\in B(H,K)$. Moreover, assume ...
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2answers
48 views

Hilbert Space - Question about norm

Let $H$ be a Hilbert space. Is it true that, if $\|x\|$ is less than or equal to $r$ and $\|y\|$ is strictly greater than $r$, then $\left\| x-\frac{ry}{\|y\|} \right\|$ is less than or equal to ...
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1answer
47 views

Functional to inner product in Hilbert triple

If $V \subset H \subset V^*$ is a Hilbert triple, and $f \in V^*$, I cannot represent $f(v) = (e,v)_V$ because we don't identify $V$ with $V^*$. But is it true that $f(v) = (e,v)_H$ for some $e$?
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1answer
167 views

Subspace of a Hilbert Space

I am sitting through a course on "Operators in Hilbert Spaces". The instructor has asked us to look at the following problem: Let $H$ be a hilbert space and $E \subset H$. E is called weakly bounded ...
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50 views

Alternative explanation for $\iint_D \left|\log \left( \frac{e}{1-z} \right) \right|^2 \ dA = \frac{\pi^3}{6}$?

I thought up a curious definite integral. Let $D = \{ z \in \mathbb{C} : |z|<1\}$. Let $A$ denote area measure on $D$, normalized so that $A(D) = \pi$. I claim that $$\iint_D \left|\log \left( ...
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0answers
67 views

If limit of $f(n)$ is zero then the operator is compact

I want to prove the following: Suppose $\mathfrak{H}$ is the Hilbertspace $l^2(\Bbb{N})$ and $T_f$ the multiplication operator on $\mathfrak{H}$, thus $T_f\psi=f\psi$ for ...
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1answer
398 views

Relation between adjoint operators/dual operators

I'm a bit confused about adjoint operators. Let $T:X \to Y$ be a linear isomorphism between Hilbert spaces. Then is it true that $(Tx,y)_Y = (x,T^*y)$ exists (does $T^*:Y \to X$ always exist)? What ...
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1answer
102 views

Connecting two Hilbert spaces' inner products via isomorphism

If I have two Hilbert spaces $X$ and $Y$ and a continuous linear isomorphism $T:X \to Y$ with continuous inverse $T^{-1}:Y\to X$, is there anyway to write $$(a,b)_X$$ as an inner product on $Y$? I ...
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0answers
82 views

Adjoint operator on subspace

Let $T:V_1 \to V_2$ be linear with adjoint $T^*:V_2^* \to V_1^*$. Suppose $V_i \subset H_i \subset V_i^*$ is a Hilbert triple. Let $f \in H_2 \subset V_2^*$. How can I interpret $T^*f$? Is it just ...
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1answer
121 views

Hilbert norm and Euclidean distance

For real matrix $X$ where $d_{i,j}^2(X)$ indicates the euclidean distance squared between the rows $i,j$ of $X$, if $d_{i,j}^2(X)=||f(X_i.)-f(X_j.)||_H$ then what would the function $f(.)$ be? Is ...
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1answer
140 views

Convex subset of Hilbert space as intersection of closed balls

How does one prove that any closed, convex, and bounded subset of a Hilbert space is the intersection of the closed balls that contain it?
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2answers
129 views

Linear operator and extension of its inverse

Let $K:H_1 \to H_2$ be a linear operator between Hilbert spaces that may not be bounded. $K$ is bounded below. So $K$ has an inverse $K^{-1}:\text{Range}(K) \to H_1$. $K^{-1}$ extends by ...