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

learn more… | top users | synonyms

5
votes
0answers
423 views

Sum of operator and adjoint is self-adjoint

In abstract Hodge theory there is the following lemma: Let $H$ be a Hilbert space and $A \in \mathcal{C}(H)$ a densely defined, closed operator (so possibly unbounded) and $A^*$ its adjoint operator. ...
2
votes
0answers
273 views

Can we construct a Hilbert space where the operator $A_u v := -\frac{1}{2} v'' + (vF + v\int_\mathbb{R} Su + u\int_\mathbb{R} Sv )'$ is symmetric?

It seems not to be a easy problem. I'd like to know if one can define a pertinent Hilbert space where the operator $$A_p v := -\frac{1}{2} v^{\prime\prime} + (vF + v\int_\mathbb{R} Sp + ...
1
vote
2answers
185 views

Under What Conditions Does the Action of the Dual Space Induce an Hermitian Inner Product?

I'm starting to learn about Dirac notation in Quantum Mechanics, and am coming from a pure background. The notes I'm reading states that we assume that the action of the dual space on the state space ...
3
votes
1answer
389 views

Estimate for Operator Norm in Hilbert Space

Let $H$ a hilbert space with an orthonormal basis $(e_n)_{n\in \mathbb{N}}$ and $F$ a linear operator, such that $\langle e_k,F e_n\rangle =:\phi(n,k)$. Find a good estimate for $\lVert F\lVert$ in ...
2
votes
1answer
164 views

Compute operator norm by image on orthonormal basis

Let $e_n$ a orthonormal basis for a Hilbert space and $T$ a bounded linear operator. Is the following correct? $$\lVert T \lVert^2 \leq \sup_{n \in \mathbb{N}} \sum_{k \in \mathbb{N}} |\langle ...
0
votes
2answers
110 views

Net of Projections

I am trouble proving the following proposition in Conway's functional analysis book. $H$ is an arbitrary Hilbert space, $I$ is an index set. Prop - Let $\{P_i:i\in I\}$ be a family of pairwise ...
3
votes
2answers
507 views

A continuity condition for a bilinear form on a Hilbert space

Let $H$ be a real Hilbert space, and let $B : H \times H \to \mathbb{R}$ be bilinear and symmetric. Suppose there is a constant $C$ such that for all $x \in H$, $|B(x,x)| \le C \|x\|^2$. Must $B$ be ...
4
votes
6answers
632 views

Bounded operator that does not attain its norm

What is a bounded operator on a Hilbert space that does not attain its norm? An example in $L^2$ or $l^2$ would be preferred. All of the simple examples I have looked at (the identity operator, the ...
2
votes
2answers
129 views

Hilbert space linear operator question

Let $\mathcal{H}$ be the vector space of all complex-valued, absolutely continuous functions on $[0,1]$ such that $f(0)=0$ and $f^{'}\in L^2[0,1]$. Define an inner product on $\mathcal{H}$ by ...
3
votes
1answer
113 views

What is $\mathcal{C}(S^{1})$? (Where $S^1$ denotes unit circle)

What is $\mathcal{C}(S^{1})$ (Continuous function on a unit circle)? (Where $S^1$ denotes unit circle) I saw this in a proof of showing Fourier Basis $S:=\{1,\sqrt{2}\cos{nx},\sqrt{2}\sin{nx}\}$ is ...
2
votes
1answer
116 views

Family of Self-Adjoint Operators that are Multiplications on a Common $L^2(\mu)$?

Suppose that $H$ is some (complex) Hilbert space and that $\{T_\alpha: \alpha \in I\}$ is some collection of bounded self-adjoint operators on $H$. A version of the spectral theorem states that for ...
2
votes
1answer
530 views

Bessel sequence in Hilbert space

I'm posting this question again because I'm still confused about the answer! A sequence $\{f_{n}\}_{n\in I}$ is called a Bessel sequence in a Hilbert space $H$, if there exists $B>0$ such that ...
0
votes
1answer
93 views

Convergence of a sequence of sets

Given a squence of sets $\{S_{n}\}_{n=1}^{\infty}$, where each $S_{n}$ is countable set. I came over this statement in some article, it says: "Let $S$ be the weak limit of $S_{n}$". But I couldn't ...
2
votes
1answer
221 views

Projections on Hilbert space

My question is: Let $H$ be a Hilbert space and $T \in B(H)$. Prove that $T$ is a projection if and only if $T$ is the identity on the orthogonal complement of its kernel. Thanks
0
votes
1answer
110 views

If a sequence is not a frame

A sequence $\{f_{n}\}_{n\in I}$ is a frame for a separable Hilbert space $H$ if there exists $0<A\leq B<\infty$ such that $$ A\|f\|^{2} \leq \sum_{n\in I}|\langle f,f_{n}\rangle|^{2}\leq ...
1
vote
2answers
979 views

Inner product on the tensor product of Hilbert spaces

Let $H_1$ and $H_2$ be Hilbert spaces with inner products $\langle\cdot,\cdot\rangle_1$ and $\langle\cdot,\cdot\rangle_2$, respectively. Then $H_1\otimes H_2$ is at least a pre-Hilbert space (we are ...
2
votes
1answer
89 views

Hilbert space question

Let $\{x_n\}$ be a sequence of pairwise orthogonal vectors in a Hilbert space $H$. Prove that the following are equivalent: a) $\displaystyle\sum_{n=1}^\infty \|x_n\|^2<\infty$ b) ...
3
votes
1answer
514 views

Bounded linear operator on a Hilbert space

I am having a bit of difficulty with the following homework problem. Let $\{x_n\}$ be an orthonormal basis in a Hilbert space $V$ over $\mathbb{C}$ and let $\{c_n\}_{n \in \mathbb{N}}$ be a fixed ...
3
votes
1answer
85 views

Function space in QM

I need to understand how one can think of a function as a vector (in Hilbert space, more specifically) so I can follow along QM texts. I've read this question's answers, but they were uninspiring to ...
0
votes
1answer
508 views

Vector space generated by the tensor products of pauli matrices

Let $\sigma_0,\sigma_x,\sigma_y,\sigma_z$ stand for the $2\times 2$ identity matrix and the well known pauli matrices: \begin{equation} ...
2
votes
1answer
105 views

A question about the proof in functional analysis

I'm now reading Pazy's book about the semi-group operator. To prove the existence of the solution of KdV equation. He define the Hilbert space $H^s(\mathbb{R})$ $$ \Vert ...
1
vote
3answers
2k views

Question on weak convergence ( Example).

Can anybody tell me why $\sin(nx)$ converges weakly in $L^2(-\pi,\pi)$. I can't see how $\sin(nx)$ can converge? Explanation with any other example will be nice as well.
0
votes
1answer
256 views

Orthonormal basis implies orthogonal basis!

If $\{\frac{f_{n}}{\|f_{n}\|}\}_{n\in I}$ is an orthonormal basis for a separable Hilbert space $H$, and $\{f_{n}\}_{n\in I}$ is a complete and orthogonal set in $H$, is it true that $\{f_{n}\}_{n\in ...
5
votes
2answers
1k views

How to show a compact, closed-range operator on an infinite-dimensional Hilbert space has finite rank, without using the open-mapping theorem?

If $H$ is an $\infty$-dimensional Hilbert space and $T:H\to{H}$ is a compact operator with closed range, how do I show that $T$ has finite rank, without using the open-mapping theorem? (The ...
2
votes
1answer
259 views

Why are only Sobolev spaces with certain exponents Hilbert Space?

I would like to know why $W^{k,2} (\Omega) $ is a Hilbert space , why is it impossible to define inner product in other Sobolev spaces, ie exponent $\ge2$ . Here $||u||_{W^{k,2} (\Omega)} $ = ...
1
vote
1answer
613 views

Poincaré inequality in unbounded domain

Help me please, how can I to show that Poincaré inequality in unbounded domain doesn't holds? Thanks a lot! If $\Omega$ is a bounded domain and $u \in H_{0}^{1}(\Omega)$ the following inequality ...
3
votes
2answers
365 views

The image of orthonormal basis under compact operator

I need a help to prove that statement: if $\{e_n\}$ an orthonormal basis in Hilbert space $H$ and $A$ is a compact operator from $H$ to $H$, then $Ae_n\rightarrow 0$. Thx for any help.
0
votes
1answer
473 views

Square root of compact operator

I'm trying to solve a functional analysis problem A self-adjoint non-negative operator $A$ on a Hilbert space $H$ is compact if and only if its $\sqrt{A}$ is compact.
2
votes
0answers
126 views

Is inversion sequentially continuous in SOT?

Let $A_n \overset{SOT}{\to} A$ where $A$ is invertible. Does $A_n^{-1} \overset{SOT}{\to} A^{-1}$? Does $A_n^{-1} \overset{WOT}{\to} A^{-1}$? EDIT: Forgot to mention $\{A,A_n\}\in\mathscr{B(H)}$ ...
3
votes
2answers
130 views

closedness of image of closed, unbounded operator

I want to prove the following: Suppose $\mathcal{H}_1$ and $\mathcal{H_2}$ are Hilbert spaces and let $T: \mathcal{D} \rightarrow \mathcal{H}_2$ be a closed operator, where $\mathcal{D} \subset ...
3
votes
1answer
389 views

Derivative of Convex Functional

Suppose that $H$ is a real Hilbert space and that $f:H \to \mathbb{R}$ is differentiable in the Frechet sense. Then we can think of the derivative as a function $f': H \to H^* = H$. Suppose that this ...
1
vote
1answer
93 views

Changing of integration and operator

I have a question which maybe looks very simple: Let $T$ be an orthogonal projection on a Hilbert space $H$. If $g(x,u)\in H$, for all $u\in \mathbb R$, and the inner product is defined by $$\langle ...
3
votes
3answers
384 views

Maximal Value of Integral

Calculate the maximal value of $\int_{-1}^1g(x)x^3 \, \mathrm{d}x$, where $g$ is subject to the conditions $\int_{-1}^1g(x)\, \mathrm{d}x = 0;\;\;\;\;\;\;\;$ $\int_{-1}^1g(x)x^2\, \mathrm{d}x = ...
1
vote
1answer
361 views

Compact Self Adjoint operator on a Hilbert Space

Let $T$ be a continuous and bounded self-adjoint compact operator on a Hilbert space H. I want to prove that if $T^2=0$, then $T=0$. Is there any thing wrong with the following: $T^2$ = $TT^*=0$ ...
1
vote
1answer
158 views

Checking axioms for inner product

I'm going through a question checking that an inner product satisfies the inner product axioms. I have a Hilbert space $H=C[-1,1]$ and for $f,g\in H$ the inner product is defined as $$\langle ...
4
votes
3answers
91 views

$\|\cdot\|_1 \leq \|\cdot\|_2 \leq \|\cdot\|_{\infty}$ for functions in $C([0,1])$?

Why does the following hold for continuous functions on $[0,1]$? $\|\cdot\|_1 \leq \|\cdot\|_2 \leq \|\cdot\|_{\infty}$
2
votes
3answers
417 views

$C[0,1]$ is NOT a Banach Space w.r.t $\|\cdot\|_2$

I'm trying to find a cauchy sequence in $C[0,1]$ that converges under $\|\cdot\|_2$ to a limit which isn't continuous. Any ideas?
0
votes
1answer
994 views

Operator Norm and Hilbert-Schmidt

I am trying to prove $\|T\| \leq \|T\|_{HS}$ I understand everything up until the following two lines, could somebody please explain why $\| Tx \| \leq \|T \|_{HS}$ $\|x\|$ implies that $\| T ...
1
vote
1answer
466 views

Proof by induction of triangle inequality in Hilbert space.

I've made proof by induction over $n$ for triangle inequality : $\left \| x+y \right \|_{e}\leq \left \| x \right \|_{e}+\left \| y \right \|_{e}$ ,where $\left \| x \right ...
2
votes
0answers
282 views

Concept of Hilbert triple

I am trying to understand the Hilbert triple $V \subset H \subset V^*$, where $V$ and $H$ are Hilbert spaces and the star denotes the dual space. Eg: $H^1 \subset L^2 \subset H^{-1}.$ If $V \subset ...
1
vote
1answer
525 views

Hilbert Spaces and Closed Subspaces

Let $H$ be a Hilbert Space, and $M$ a closed subspace. Is it true that $H = M \bigoplus M^{\perp}$ Does this hold if $M$ is not closed? Or only if $H$ is finite/infinite dimensional?
1
vote
1answer
101 views

Prove that the space is not complete

Let $X$ be a separable space with infinite dimension, let $(\cdot,\cdot)$ and $\|\cdot \|$ be the scalar product and the norm of $X$, and $\{e_n\}_n$ be an orthonormal basis of $X$. We define ...
3
votes
0answers
168 views

Is there a deeper connection between the two Riesz's Representation Theorems?

I have been reading Kreyszig's Functional Analysis when I encountered two versions of Riesz's Representation Theorems: (1) Every bounded linear functional $f$ on a Hilbert space $H$ can be ...
2
votes
1answer
199 views

Operator Norm $\| T\|$

Would somebody mind explaining why if $T$ is a continuous and bounded operator on a Hilbert space $H$, we have $$\|T\| = 1 \;\;\;\Rightarrow \;\;\;\|Te_n \| = \|e_n\|\;\;\mbox{for all }\;\;x\in H$$ ...
2
votes
2answers
164 views

Is $L^2(D)$ separable?

Let $D$ be a bounded connected open subset of $R^n$ and $μ$ is a finite measure on $D$, say the Lebesgue measure. Is $L_2(μ)$ separable? Is a bounded sequence $\{f_k\}$ of $L^2(μ)$ pre-compact?
4
votes
1answer
445 views

Weak convergence

Let $H$ be a Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and let $V,W$ be two closed subspaces. For $x_0\in H$ we may define the sequence of projections $$x_{2n+1}=P_W(x_{2n}), \qquad ...
1
vote
2answers
348 views

Hilbert space operators, relation between trace, rank and range

If $A\colon H\to S$ is a bounded operator on a Hilbert space $H$, and $S\subset H$. It is known that $\operatorname{trace}(A)=\sum_{n} \langle Af_n,f_n\rangle$ for any orthonormal basis $\{f_{n}\}$. ...
5
votes
2answers
173 views

$L^{2}$ functions

Let $f(x)$ be a continuous function for all $x\in \mathbb R$, such that $f\in L^{2}(\mathbb R)$ (i.e., $\int_{-\infty}^{\infty}|f(x)|^{2}dx<\infty$), and define $$f_{o}(x):=\sup_{|x-y|\leq ...
8
votes
1answer
874 views

Is the right shift operator bounded?

I was reading my lecture notes for functional analysis when I came across the following statement: Let $(e_{n})$ be a total orthonormal sequence in a separable Hilbert space H. The right shift ...
1
vote
1answer
405 views

The exponent of self-adjoint operator

If $X$ is a Hilbert space and $A$ is an unbounded self-adjoint operator on $X$, is it necessarily that $A^2$ is self-adjoint as well?(admittedly, $A^2$ is densely defined)