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

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1answer
10 views

Projection with modulus less than one

Let $X$ be an Hilbert Space, $X=Y\bigoplus Z$ where $Y$, $Z$ are both closed subspaces. Let $P:X \rightarrow X$ $P(y+z)= y$ be the canonical projection, then $||P|| \leq 1 \implies Y=Z^{\bot}$ ...
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1answer
38 views

Orthogonal complement properties in Hilbert spaces

We have a Hilbert space $H$ and $x_0\in H$. We need to show that if we let $V$ be a closed subspace of $H$, then $$ \min\{\|x-x_0\|\,:\,x\in V\}=\max\{|\langle y,x_0\rangle|\,:\,y\in ...
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2answers
44 views

showing $\inf \sigma (T) \leq \mu \leq \sup \sigma (T)$, where $\mu \in V(T)$

I am trying to prove the following: Let $H$ be a Hilbert space, and $T\in B(H)$ be a self-adjoint operator. Then for all $\mu \in V(T)$, $\inf \left\{\lambda: \lambda \in \sigma (T) \right\}\leq \mu ...
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1answer
29 views

An identity on direct sum of Hilbert spaces

Let $M_i$ are the set of smooth complex valued functions ($i=0,1,2,...$) $L^2(M_i)$ are Hilbert spaces on $M_i$ then can we say $$L^2(\bigoplus_{i=0}^\infty M_i)\cong \bigoplus_{i=0}^\infty ...
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0answers
36 views

Question about Morse index

in general the Morse index of a critical point $p$ is the suprimum of the dimensions of sub spaces where $f''(p)$ is negative definite but whene $f''(p)=I-T$ ($f''(p)$ is a compact perturbation of ...
2
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1answer
25 views

Unitary transformation between complete and orthonormal bases

I'm using the Dirac notation for vectors here, since this is a quantum mechanics question. Suppose the complete orthonormal bases $\{|\psi_n\rangle\}$ and $\{|\psi{'}_n\rangle\}$ are related by the ...
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0answers
60 views

Expectation of $p$-norm under a Gaussian on the Hilbert space $L^2(S^1)$

Let $\mu$ be a centered Gaussian measure with (nondegenerate) covariance $Q$ on the Hilbert space $L^2(S^1;\mathbb R)$ where $S^1$ is the circle. We can take for example the covariance ...
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1answer
19 views

equivalency of weak convergence and strong convergence for this family of sequences

Let $H$ be a Hilbert space and let $f_n \in H$ be a sequence of orthogonal elements i.e $<f_n,f_m>=0 $ if $n\ne m$. Define the element $F_N= f_1 + f_2 +...+ f_N$ for each $ N\in \mathbb N$. ...
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2answers
69 views

Dense Countable basis on Hilbert space

Let say that I have a $H$ hilbert space and linear independent countable set $\beta =\{ \beta_1 , \beta_2, \beta_3... \}$ such that $span(\beta)$ is dense set in H. does $span(\beta-\beta_1) =span( ...
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1answer
78 views

Hilbert space $L^{2}(0,\pi)$

I wanted to know how I should proceed if I wanted to prove that the closed subspace of $L^{2}(0,\pi)$ generated by {$\sin(kx): k=1,2,...$} coincides with $L^{2}(0,\pi)$. Thanks.
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0answers
42 views

Generalisation of Gramian determinant

i'm wondering about those facts of basic linear algebra: if you have $n$ vectors $x_1,...,x_n \in \mathbb{R}^n$, you can easily test their linear dependance by computing their Gramian Matrix $M$ whose ...
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0answers
35 views

Domains of operators defined by quadratic forms

Consider a separable Hilbert space $H$. Say we have two lower-bounded, densely defined quadratic forms $a$ and $b$ with respective domains $D[a],D[b] \subset H$ such that $D[b] \subset D[a]$ ...
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1answer
53 views

Composing Projections on a Hilbert Space

Let $P,Q$ be projections on a Hilbert space such that $PQ$ is a projection. I have been able to prove that $PQ=QP$. I want to show that $ker(PQ)$ is contained in $ker(P)+ker(Q)$. If there's a ...
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0answers
41 views

Subspace of Hilbert space

I met a problem related with proving whether a subspace of Hilbert space is still a Hilbert space. Given the space $V = \{v \in H^{1}(0,1), \int_0^1 v \mathrm{d} x = 0 \}$, in which $H^{1} (0,1)$ is a ...
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1answer
25 views

Is tensor product commutative on orthonormal basis?

In general the tensor product $\varphi\otimes\psi$ is not commutative, but I was thinking that if I have tensor product on two orthonormal bases of Hilbert spaces are they commutative i.e is ...
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1answer
72 views

Poincaré inequality for a subspace of $H^2(\Omega)$

Suppose that $\Omega\subset\mathbb{R}^d$ is a smooth, bounded, and connected domain. Let \begin{equation} H=\{u\in H^2(\Omega):\int_\Omega u(x) dx=0 ~\text{and}~ \nabla u\cdot v=0~ ...
2
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1answer
40 views

Density result in Hilbert space

Assume that $b\in \mathbb{C}$ such that $0<\vert b \vert <1$. We consider the familly $f_{p}=\{1,b^{p},b^{2p},b^{3p},b^{4p},...,b^{np},...)$. How can one prove that $\operatorname{Span}(f_{p}, \ ...
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1answer
62 views

Inner product on direct sum of Hilbert spaces

Let $H_1$ and $H_2$ are two different Hilbert spaces then how can we define the inner product on $H_1\oplus H_2$
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1answer
68 views

how to prove this density result?

Assume that $b\in \mathbb{C}$ such that $0<\vert b \vert <1$. We consider the familly $f_{p}=\{1,b^{p},b^{2p},b^{3p},b^{4p},...,b^{np},...)$. How can one prove that $\operatorname{Span}(f_{p}, \ ...
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2answers
67 views

Hilbert vs Inner Product Space

What is the difference between a Hilbert space and an Inner Product space? They both seem to be defined as simply a vector space equipped with an inner product. Also can a metric always be defined ...
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1answer
74 views

Is the space of continuous functions a Cauchy complete?

I am so new to functional analysis so I am looking for an answer of a confusion I am having right now in my mind because I have seen many different answers for the question I am gonna ask below. I ...
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3answers
93 views

polar decomposition proof

Let $H$ be a hilbert space and $T$ a bounded linear operator on $H$. I'm trying to prove that there is a partial isometry $V$ on the closure of $Im(|T|)$ such that $T=V|T|$ and $|T|=V^*T$, where ...
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2answers
40 views

Orthogonal representation of finite operator

I would like to know if my proof is correct. Statement: Let $T$ be a finite rank operator on a Hilbert space $\mathscr{H}$. Show that $\forall \, h \, \in \mathscr{H}, \, T(h)$ can be written as ...
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0answers
65 views

Proving that a positive operator has a unique square root

Rudin's functional analysis page 331 theorem 12.33) proves this. He proves uniqueness by 'going back' to the general algebra setting. I was just wondering whether there is a more direct way of doing ...
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0answers
39 views

question on Hilbert space [duplicate]

I know that $L^{2}(\mathbb R)$ is an Hilbert space. I am wondering how to prove that it is the only one? that is how to prove that $L^{p}(\mathbb R)$ spaces are not Hilbert spaces for $p \ne 2$? ...
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2answers
36 views

Showing $P^2=P$ for $P(v)=\frac{\langle v,w\rangle }{||w||^2}w$

My book asserts that for fixed $w$ where $w\neq 0$ that $P^2=P$ for $P(v)=\frac{\langle v,w\rangle }{||w||^2}w$ My book has a general corralary that $v\to P(v)$ is a bounded linear transformation and ...
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2answers
68 views

uniqueness of positive operator

Let $A,B$ be commuting positive operators on a hilbert space such that $\langle(A-B)(A+B)x,x\rangle=0$ for all $x$ in the hilbert space. Prove that $A=B$. My attempt: The above implies that $A=B$ on ...
2
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1answer
106 views

$\langle Tx,x \rangle=0$ proof

If $T$ is a bounded operator on a hilbert space $H$ and $\langle Tx,x \rangle=0$ for all $x$ in $H$, then $T=0$. I'm considering what we can conclude if $\langle Tx,x \rangle=0$ for all $x$ in some ...
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0answers
40 views

Lack of a polar decomposition

Prove that the left and right shifts on $l_{2}$ have no polar decomposition (i.e. $UP$ where $U$ is unitary and $P$ is positive).
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2answers
56 views

Rudin Polar decomposition

On page 332 theorem 12.35b) of Rudin functional analysis is show that if T is normal then it has a polar decomposition $T=UP$. Does he mean that $P=|T|$? He's a bit ambiguous as to how he defines ...
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2answers
75 views

functional calculus and spectral measure

Let $T$ be a normal operator and $f$ be a bounded borel function on ${\sigma}(T)$. If $E_{T}$ and $E_{f(T)}$ are the spectral decompositions of $T$ and $f(T)$ respectively, prove that for any borel ...
0
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2answers
85 views

generalized eigenspace direct sum

Similar to the way an infinite dimensional hilbert space can be written as a direct sum of eigenspaces of a normal compact operator, I was wondering whether it can be written as a direct sum of ...
2
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1answer
75 views

Functional weakly lower-semicontinuous [duplicate]

If $X$ is a topological space, then a functional $\varphi:X\to\mathbb{R}$ is lower-semicontinuous (l.s.c) if $\varphi^{-1}(a,\infty)$ is open in $X$ for any $a\in\mathbb{R}$. If $X$ is a Hilbert ...
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1answer
52 views

Dual of Hilbert space dense in dual of Reflexive space.

I don't see how to solve this problem which I think should be easy: Let Y be a reflexive space. Assume $Y$ is continuously embedded in a Hilbert space $H$ and $Y$ is dense in $H$. Show that $H^*$ is ...
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1answer
36 views

spectral measure and integral query

I have proved the 'resolution of the identity' for a normal operator, namely that there is a unique spectral measure E such that $\int_{{\sigma}(T)} {\lambda}\,dE=T$ If (${\lambda}_{n}$) is the ...
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3answers
138 views

Borel functional calculus

For a normal operator T, we have a resolution of the identity $\int_{{\sigma}(T)} {\lambda}\,dE=T$. If $T$ is in addition compact , we have that $\sum_{n=1}^{{\infty}}{\lambda}_{n}\langle ...
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1answer
75 views

Projection operator in Hilbert space

Let $H$ be a Hilbert space, can we find an increasing net of finite rank projections which converge to the identity in the strong operator topology? And I think if $H$ is separable, we can find an ...
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0answers
25 views

The Haar basis ,proof of orthonoramality.

please i have this problem and i known how to prove completeness but do not know how to prove that it is orthonormal. I will appreciate it if anyone can help me. Given that $n\geq1$ write ...
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1answer
44 views

Fredholm operators in Hilbert spaces

Suppose $T_r$ and $T_l$ are the left and the right translations in $l_2$. $T_l$ maps $(x_1,x_2,x_3,...)$ to $(x_2,x_3,x_4,...)$, $T_r$ maps $(x_1,x_2,x_3,...)$ to $(0,x_1,x_2,...)$. It can be easily ...
4
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1answer
86 views

Spectral decomposition of normal operator

Define $T$ from $L_{2}(R)$ into itself by $T(f)(t)=f(t+1)$. Show that $T$ is normal and finds its spectral decomposition. I've shown that $f$ is normal (in fact it's unitary) but how do I find its ...
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2answers
103 views

Cauchy-Schwarz Inequality by sum of squares.

i'm trying to solve: Prove the Cauchy-Schwarz inequality by writing $||x||^2||y||^2− |⟨x,y⟩|^2$ as a sum of squares. I'm fairly well versed in Cauchy Schwarz and know several proofs but I'm confused ...
0
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1answer
18 views

Identification of tensor product spaces

Let $P_1, P_2$ be (probability) measures, $\Omega_1, \Omega_2 \subset \mathbb{R}^n$ . Prove that $L_{P_1 \otimes P_2}^2(\Omega_1 \times \Omega_2)$ and $L_{P_1}^2(\Omega_1) \otimes L_{P_2}^2(\Omega_2)$ ...
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0answers
11 views

Universal property of tensor products of real Hilbert spaces

I have the following exercise where I could need some hints: Let H1, H2 be real Hilbert spaces. Prove that there is a weak Hilbert-Schmidt mapping $$ p: H_1 \times H_2 \rightarrow H_1 \otimes H_2 ...
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1answer
52 views

Polar decomposition corollary

Let $T$ be a compact operator on an infinite dimensional Hilbert space. Let $|T|=(T^*T)^{0.5}$. By the polar decomposition theorem there is a partial isometry $S$ of the closure of Im$(|T|)$ such that ...
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0answers
84 views

Duality pairing and difference with inner product in Hilbert spaces

My question is an extension to the post Acting of a dual pairing in Sobolev Spaces. Here duality pairings were discussed and even given explicit examples. Let $U$ and $V$ be Hilbert spaces such that ...
4
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0answers
100 views

Spectral decomposition of $TT^*$

On $l_{2}$ let $T$ be given by $Te_{n}=\frac{e_{n+1}}{n+1}$ where $(e_{n})_{n\ge1}$ is the canonical orthonormal basis. Find the spectral decomposition of $TT^*$. I find that ...
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0answers
55 views

about a theorem of weakly lower semicontinuous functions

I am studying the proof of the following theorem Theorem: Let $E$ a Hilbert space and suppose that $\varphi :E \rightarrow R$ is a weakly lower semicontinuous functional. Suppose that $\varphi$ is ...
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78 views

Does this sequence of operators converge in norm or strongly?

Let $H$ be a Hilbert space and $\mathcal{L}(H)$ the set of all bounded linear operators $L:H\to H$, equiped with the usual norm $\|\cdot\|_{\mathcal{L}}$. Let $T:D(T)\subset H\to H$ be a ...
2
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2answers
66 views

Has this theorem (on existence of inverse) an analogous for unbounded operators?

Let $S,T:X\to X$ be bounded linear operators, where $X$ is a Banach space. It's a consequence of the Banach Fixed Point Theorem that if $T$ is invertible and $\|T-S\|\|T^{-1}\|<1$ than $S$ is ...
0
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1answer
14 views

Image and Kernel of a certain bounded operator

Consider and Hilbert Space $X$, $T\in B(X)$ and a scalar $\mu$ s.t. $|\mu|=||T||$ By a simple argument I deduced that $\ker(\mu I- T)=\ker(\bar\mu I-T^*$) where $*$ denotes the adjoint. I am then ...