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

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Hilbert space without the projection theorem

One succinct statement of the projection theorem in Hilbert space is $A+A^\bot=\scr H$, where $A\in\scr C$, the set of closed subspaces of $\scr H$. (We will also denote the set of all subspaces by ...
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66 views

Prove that if $X$ is a Hilbert space, then $B(X)$ is not a Hilbert space

I`m having a homework question that goes like this: X is a Hilbert space, a complete inner product space, show that B(X) is not a Hilbert space. I`m quite stuck and I would love to understand this ...
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123 views

Nonseparable $L^2$ space built on a sigma finite measure space

Is it possible to have a nonseparable $L^2$ Hilbert space for which the underlying measure space is sigma finite? I appreciate any example but prefer one built on the Borel sigma algebra of some ...
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1answer
74 views

I dont understand this notation

I`m having a homework question that goes like this: $X$ is a Hilbert space, a complete inner product space, show that $B(X)$ is not a Hilbert space. My only question for now is what does $B(X)$ ...
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47 views

Inner product space or Hilbert space of Quaternionic Functions

In what ways can you define an inner product, $<f,g>$, to create an inner product space or Hilbert space on the set of quaternionic functions $f:\mathbb{H} \rightarrow \mathbb{H}$?
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2k views

Operator norm of orthogonal projection

I was assigned the following homework problem: "Let $P:\mathcal{H} \to \mathcal{H}$ be bounded and linear. Assume it satisfies $P^2 = P$ and $P^\star = P$. Show $\|P\| \le 1$." This isn't too hard ...
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52 views

The subspace sum of closed subspaces is closed [duplicate]

Given an arbitrary Hilbert space $\scr H$ and closed subspaces $A,B\subseteq\scr H$ with trivial intersection, is it true that $A+B=\{x+y:x\in A,y\in B\}$ is closed? So far, I have the following: Let ...
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1answer
144 views

Strengthened Cauchy-Schwarz inequality

I'm looking for some simple proof of the following consequence of the "strengthened" Cauchy-Schwarz inequality: Let $\mathcal{H}$ be a real Hilbert space such that ...
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2answers
75 views

Find norm of operator

I have a linear functional $$A: L_2[0,2] \to \mathbb R, Ax = \int_0^2(t^2+2)x(t)dt$$ I need to find $C$, trying to measure $C$ and $||Ax||$ to find it, but how can I do it in this problem?
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1answer
39 views

Series convergence in Hilbert space and dual.

I'd like to prove that: $$ \|u_\varepsilon-f\|_*\rightarrow0 \quad\text{in }V^* $$ with $V$ Hilbert and $V^*$ its dual. In particular $u_\varepsilon\in V$. From the precedent points of the proof I ...
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37 views

Showing that $R(T)=R(T^*)$ for a normal operator $T$

For a normal operator $T$ acting on a Hilbert space it is easy to show that the kernel of $T$ coincides with the kernel of the adjoint $T^*$. Thus the norm-closures of the ranges $R(T)$ and $R(T^*)$ ...
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71 views

Necessity of hypothesis in distance from a set in an inner product space

In Kreyzig's Functional Analysis book, one theorem in inner product spaces is about the existence and uniqueness of a minimal point from a set. 3.3-1 Theorem (Minimizing vector). Let $X$ be an ...
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1answer
34 views

Increasing convex-like function in Hilbert space

I am intersted with the differential equation $$x'(t)=f(t,x(t)),\ t\in \mathbb{R}.$$ Can we find an example of a Hilbert space $H$ and a function $f:\mathbb{R}\times H \to H$ which satisfy the two ...
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1answer
92 views

Weak and Norm convergence in Banach Space

I know (and have proven) that in a Hilbert space, $\mathscr{H}$, if a sequence $z_i\overset{w}{\to}z$ and $\|z_i\|\to\|z\|$, then $\|z_i-z\|\to0$. I'm trying to find a counterexample in a Banach ...
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1answer
38 views

How would you define the square of the linear operator

If you define the linear operator norm of $A:X\to Y$ to be $$\|A\|_{op} = \inf\{C>0: \|Ax\|_Y \leq C\|x\|_X \text{ for all } x \in X \}$$ Then how would you define $\|A\|_{op}^2$? My guess is you ...
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2answers
182 views

Want to show that $\{e^{i2\pi nx}:n\in\mathbb Z\}$ form an orthonormal basis for 1-periodic $L^2$ functions.

So here is my problem, I would like to prove that $\{e^{i2\pi nx}:n\in\mathbb Z\}$ form an orthonormal basis for 1- periodic $L^2([0,1])$ functions with respect to, $$\langle ...
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78 views

Proving the completeness of $\mathcal{L}(\mathcal{H})$

Here $\mathcal{L}(\mathcal{H})$ denotes the vector space of all bounded linear operators on a Hilbert space $\mathcal{H}$. We can define a norm on $\mathcal{L}(\mathcal{H})$ by $\|T\| = \inf\{B : ...
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46 views

On a Variational Inequality

Let $H$ be a Hilbert space with real inner product. Consider $f: C \rightarrow C$, where $C \subset H$ is closed and convex. I am not sure about the variational inequality problem: find $x \in H$ ...
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1answer
435 views

Prove vectorspace of bounded functions with supremum-norm is complete and no hilbert space

I have the following: Consider the real vectorspace with bounded functions $$V = \{f:[0,1]\rightarrow\mathbb{R} | \exists C > 0 : f([0,1])\subset[-C,C]\}$$ and the supremum-norm $$||f||_\infty ...
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56 views

$\left\{x\in H: 2\leq \|x\|\leq 5\right\}$ is compact?

In a Hilbert space $H$ of dimention infinite, $A=\left\{x\in H:2\leq \|x\|\leq 5\right\}$ is compact? (totally bounded and complete) Thanks in advance.
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81 views

Bounded Sesquilinear form

Let $X$ and $Y$ be normed spaces. Show that a bounded sesquilinear form $h$ on $X \times Y$ is jointly continuous in both variables.
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46 views

Want to prove an inequality of two norms in a Hilbert space

So here is my problem, Let $D:=[-d,d]\times[-d,d]$ and $C_0^{\infty}$(D) be the set of all smooth functions with compact support in $D$ which are zero on the boundary of $D$. Moreover we have the ...
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1answer
51 views

product of Hilbert spaces

Let $H$ be an infinite dimensional Hilbert space. claim: $H\times H$ with the norm $\|(x,y)\|=\|x\|+\|y\|$ is an Hilbert space. I can't find a counterexample..
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27 views

Easy exercise operators on Hilbert space

Let $(H,\langle\,|\,\rangle)$ be a separable Hilbert space on $\mathbb{C}$. $\rho_{\psi}:=\langle\psi,\,\rangle\psi$, where $\psi\in H$ is such that $\|\psi\|=1$. ...
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44 views

Orthogonal Projector

Let $(H,\langle\,|\,\rangle)$ be a separable Hilbert space on $\mathbb{C}$. $P_{\psi}:=\langle\psi,\,\rangle\psi$, where $\psi\in H$ is such that $\|\psi\|=1$. I have to prove that $P_{\psi}$ is an ...
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109 views

Unitary operator on dense set, Unique extension?

given two Hilberspace $H_1$ and $H_2$. Let $V\subset H_1$ and $W\subset H_2$ be dense subspaces. Furthermore let $U: V \rightarrow W$ be an unitary operator. I just want to know whether there is a ...
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34 views

Limit of function of an operator

Let $A_n$ be a sequence of bounded, self-adjoint operators on Hilbert space $\mathcal{H}$. Let us assume that for some vector $\psi\in\mathcal{H}$, $$\lim_{n\rightarrow\infty}A_n\psi = \alpha ...
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1answer
63 views

isometric embedding of l^2

CLAIM: Let $H$ be an infinite dimensional $\mathbb{R}$-Hilbert space. Then the $\ell^2$ sequence space can be embedded in $H$. I think it could be true since every Hilbert space has an orthonormal ...
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30 views

Bounding the distance between $L_\infty$ and $L_2$ for a continuous function

Consider a set of continuous (or even differentiable) functions $f_i(x)$, all defined for $x\in [a,b]$ for $i=1\ldots,N$. Can one define a uniform constant $c$ (which may depend on $f$) such that ...
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1answer
147 views

Is every closed set $K\subseteq \mathbb{C}$ the essential range of a measurable function?

For a complex-valued function $h$ on a measure space $(S,\Sigma, \mu)$, the $\textit{essential range}$ of $h$ is the set of all $\lambda \in \mathbb{C}$ such that for all $\epsilon >0$ the ...
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1answer
304 views

Range and kernel of linear operators

I have a compact linear operator $T$, and I would like to show $$\operatorname{range}(\lambda I-T)=(\ker(\overline{\lambda}I-T^*))^\perp.$$ I have shown the forward inclusion "$\subset$" directly by ...
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2answers
458 views

Compact operator whose range is not closed

I am asked to find a compact operator (on a Hilbert space) whose range is not closed, but I am having trouble coming up with one. My guess is that you need to have some sequence in the range that ...
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1answer
65 views

Vanishing of a quadratic form along the orbits of a unitary group

Let $H$ be a (complex) Hilbert space and let $B\colon H\times H\to \mathbb{R}$ be a continuous sesquilinear form (i.e. a continuous function that is linear in one argument and conjugate-linear in the ...
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1answer
41 views

A question about orthogonal projection in Hilbert spaces

Let S be a closed linear subspace of a Hilbert space $H$, and $P_S$ the associated orthogonal projection. I need to verify the following properties. i) $||P_S(x)||\le||x||$ ii)$P_{S^\perp}=I-P_S$, ...
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1answer
37 views

Why is oblique projection not a self adjoint operator?

Why is oblique projection not a self adjoint operator? Here is an explanation of oblique projection.
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1answer
83 views

Finding an orthonormal basis from an existing one in a Hilbert space

Suppose we are given a separable Hilbert space $H$ with countable orthonormal basis $\{e_n\}$. Suppose we are given an orthonormal set $\{f_n\}$ such that $\sum\|e_n-f_n\| < 1$. How do we prove ...
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2answers
86 views

Convergence of sums using Hilbert space techniques [duplicate]

Let $a_n$ be a sequence of real numbers such that $\sum_{n=1}^{\infty} a_nb_n < \infty$ for any sequence $b_n$ satisfying $\sum_{n=1}^{\infty}b_n^2 < \infty$. Prove that ...
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1answer
97 views

Weak convergence of subsequence in Hilbert spaces

Prove that if $x_n$ is a sequence in $H$ (Hilbert space) with $\sup_n||x_n||\le1$, then there is a subsequence $\{x_{n_j}\}$ and an element $x$ of $H$ with $||x||\le 1$ such that $x_{n_j}$ converges ...
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2answers
69 views

Sequence of unit vectors in a Hilbert space

Question: Let $H$ be a Hilbert space and $\{\xi_{i}\}\subset H$ be a sequence of unit vectors. Suppose that $||T_{j}(\xi_{i})-\xi_{i}||\rightarrow0$ as $i\rightarrow\infty$, for $j=1, 2, ...n$ (here ...
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1answer
58 views

Show that a Hilbert space with two inner products has a basis that is orthogonal with respect to both inner products

Let $\mathcal{H}$ be a complex, $n$-dimensional Hilbert space with two inner products $\langle \cdot, \cdot \rangle_1$, $\langle \cdot, \cdot \rangle_2$. Show that there exists a basis $ X = x_1, ...
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2answers
109 views

showing uniqueness of a Hahn Banach extension

I am trying to prove the following: If $H$ is a Hilbert space and $G\subseteq H$ is a closed linear subspace, then any bounded linear functional on $G$ has a unique Hahn-Banach extension on $H$. So ...
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1answer
134 views

Proof of an identity in a Hilbert space

Let $x_1,x_2\in H$ be two unit vectors in a Hilbert space $H$ and let $t_1,t_2 : B(H)\rightarrow\mathbb{C}$ be the linear functionals given by $t_j(a) =\langle ax_j,x_j\rangle$. Define ...
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1answer
87 views

Finding the Hilbert Adjoint in this case

If we let $H$ be a Hilbert space with inner product $\langle.,.\rangle$. And we fix $y, z \in H$. Then let $T:H\rightarrow H$ be the bounded linear operator $Tx = \langle x,y\rangle z$. Then what is ...
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1answer
99 views

Compact operator on invariant subspace is compact

Statement: Let $T \in \mathscr{B}(\mathscr{H})$, where $T$ is a compact operator. Let $M$ be a closed invariant subspace of $T$. Show that the restriction of $T$ to $M$ is compact. Attempted Proof: ...
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1answer
37 views

Dimension of intertwining space of unitary representation

I'm currently trying to read through an article by Poguntke, to be found here. The main theorem of the article is the following: Theorem. Let $\pi$ and $\pi'$ be irreducible unitary ...
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39 views

Linear map in Hilbert space.

If you have a linear map $h\mapsto T(h)$ from $H_1$ a real separable space, to Hilbert space $H_2$, it seem that this maps provides an isometry of $H_1$ onto a closed subspace of $H_2$. I try to ...
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1answer
341 views

functions orthogonal to the exponential Bell polynomials

Consider the single variable Bell polynomials $\phi_{n}(x)$ given by: $$\phi_{n}(x)=e^{-x}\sum_{k=0}^{\infty}\frac{k^{n}x^{k}}{k!}$$ I am looking for a set of functions $\tilde{\phi}_{n}(x)$ such ...
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2answers
67 views

Is it Hermitian or not?

I have a question, since I'm realy consfused. I am doing quantum information theorem, and there's a theorem that says, that for Hermitian $A\in L(H)$ ($H$ some finite dimensional Hilbertspace) then ...
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1answer
35 views

Let $x\in C^1[-\pi,\pi].$ Prove that $|\int_{-\pi}^{\pi}(x(t).cost-x'(t).sint)dt|^2\le 2\pi\int_{-\pi}^{\pi}(|x(t)|^2+|x'(t)|^2)dt$

Let $x\in C^1[-\pi,\pi].$ Prove that $|\int_{-\pi}^{\pi}(x(t).cost-x'(t).sint)dt|^2\le 2\pi\int_{-\pi}^{\pi}(|x(t)|^2+|x'(t)|^2)dt$ We have no idea to approach the poroblem. Please help??
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105 views

Two questions on Banach-valued spaces of integrable functions

Let $V$ be a Banach space with dual $V'$ and suppose that $V$ is included into the Hilbert space $H$ so that the inclusion is continuous and dense. Then after identification $H = H'%$ we have $V ...