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

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1answer
292 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 ...
2
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1answer
270 views

A generalization of the Cauchy-Schwarz inequality to linear operators

If $A$ is an operator and $A \in \mathcal{B_{+}(X)}$ (the set of the positive operators) then the generalization of the Cauchy-Buniakowsky-Schwarz inequality holds: $$|\langle Ax,y\rangle| \leq ...
2
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1answer
1k views

Orthogonal projection on the Hilbert space .

I want to prove the following: If $X$ is a Hilbert space and $Y$ is a closed subspace of $X$, then every $x\in X$ can be written as $x=y+z $ where $y\in Y$, $z \in Y^\perp$. The ...
2
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1answer
62 views

Finding the minimizing vector of a $l_{2}$ sequence

I am working on a problem sheet and this question has me stuck. A little guidance will be appreciated. Let $X = l_{2}$. Let $x \in X$ be given by $x = \{\frac{1}{2^{i}} \}^{\infty}_{i=1}$ Let $M ...
2
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1answer
232 views

Compact operator on $l^2$

Let A be a bounded linear operator on $l^2$ defined by A($a_n$)=($\frac{1}{n} a_n$). Would you help me to prove that A is compact operator. I guess the answer using an approximation by a sequences of ...
2
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2answers
124 views

Theorem about orthogonal system in inner product space.

It is known that "If $\{x_n\}$ is a sequence in a real Hilbert space $H$ satisfying $$ \langle x_n, x_m\rangle =0 \quad\forall n\ne m, $$ then $\displaystyle\sum_{n=1}^{\infty}x_n$ is convergent if ...
2
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1answer
155 views

Equivalents norms in Sobolev Spaces

I know that this is classical but I have never do the calculations to show that the norms in the sobolev space $W^{k,p}(\Omega)$ \begin{equation} \|u\|_{k,p,\Omega}= \Bigl(\int_{\Omega} ...
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2answers
120 views

proving “$C^1([−1,1])$ is dense in the given space with given norm”

Define $$E = \left \{ f \in W^{1,2} (-1,1) \; | \; \| f \|_E := \left( \int_{-1}^1 (1-x^2 ) | f' (x) |^2 dx + \int_{-1}^1 | f(x) |^2 dx \right)^{\frac{1}{2}} < \infty \right \}.$$ Then how can I ...
2
votes
2answers
98 views

Minimizing a functional on $L^2$

Let $$ \mathcal{M} := \left\{f \in L^2([0,\pi]): \int_0^\pi f(x)\cos x dx = \int_0^\pi f(x)\sin x dx = 1\right\}. $$ Solve this problem: $$ \tag{P} \min_{\mathcal M} \int_0^\pi ...
2
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1answer
471 views

Sesquilinear forms on Hilbert spaces

Definition of the problem Let $\mathcal{H}$ be a Hilbert space, and let $B:\mathcal{H}\times\mathcal{H}\rightarrow\mathbb{K}$ be a sesquilinear form. Prove that TFAE: $(i)$ $B$ is continuous. ...
2
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1answer
106 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
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1answer
214 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
2
votes
1answer
218 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)} $ = ...
2
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1answer
169 views

how to show $f$ attains a minimum?

Let $H$ be a Hilbert space and let $f\colon H\rightarrow \mathbb{R}$ be a continuous convex function such that $f(x_n)\rightarrow\infty$ whenever $\lVert x_n\rVert\to\infty$. We need to show that $f$ ...
2
votes
1answer
176 views

What is the orthogonal complement of this subset of $L^2[0,1]$?

Let $A\subset[0,1]$ be measurable, and let $g\in L^2(A,dx)$. Let $C=\{f\in L^2[0,1]:m\{x\in A:f(x) \ne g(x)\}=0 \}$, that is, the set of functions which are equivalent to $g$ on $A$. Prove that $C$ ...
2
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1answer
106 views

Hilbert sum of $L_2(X_\nu,\mu_\nu)$ spaces.

Let $\{(X_\nu,\mu_\nu):\nu\in\Lambda\}$ be a family of measurable spaces. Is it true that $\bigoplus_2\{L_2(X_\nu,\mu_\nu):\nu\in\Lambda\}$ isometrically isomorphic to ...
2
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1answer
127 views

A question about projection in Hilbert space .

Let $a$ be a non-zero element of an Hilbert space $H$. I try to prove that for every $x\in H$, $$ d(x, \{a\}^{\perp})=\frac{\left|\langle x,a\rangle \right|}{\left\|a\right\|}. $$ So $d(x, ...
2
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1answer
104 views

Hilbert space on a finite set

If X is a finite set, what does the Hilbert space $L^2(X)$ means? - saw this notion on The Princeton Companion to Mathematics.
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3answers
1k views

Compact operator

If $H$ and $K$ are Hilbert spaces,show that if $T:H\longrightarrow K$ is a compact operator and $\{e_{n}\}$ is any orthonormal sequence in $H$ then $\|Te_{n}\|\to0$.Is the converse true? thanks.
2
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1answer
42 views

“Almost” Hilbert spaces

This question is a bit (very?) vague. Is there some notion of how "close" a Banach space is to being a Hilbert space? What I have in mind is something like a real or complex valued function on ...
2
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1answer
37 views

If a compact operator satisfies $T^nx\to0$ weakly for all $x$, then $\|T^n\|\to0$

Let $H$ be a real Hilbert space, $T:H\to H$ be a compact operator. Suppose that for every $x\in H$, sequence $(T^n x)_{n\in \mathbb{N}}$ converges weakly to $0$. How to prove that $ ...
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2answers
71 views

Properties of a set in $\ell^2$ space

Let $\ell^2 = \{x= (x_1,x_2,x_3,\ldots): x_n\in \mathbb C\text{ and } \sum_{n=1}^\infty |x_n|^2 < \infty\}$ and $e_n \in \ell^2 $ be the sequence whose $n$-th element is 1 and all other elements ...
2
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1answer
32 views

Weak convergence and infinite sum

Suppose that $\psi_n$ converges weakly to $\psi$ in a Hilbert space $H$. Assume further $\{\phi_k\}$ is an orthonormal sequence in $H$. Is it plausible that ...
2
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1answer
37 views

Closed subspace $M=(M^{\perp})^{\perp}$ in PRE hilbert spaces. [duplicate]

Is it true that $M=(M^{\perp})^{\perp}$ if $M$ is a closed subspace of a PRE hilbert space (a space with a scalar product, but that is not complete)? The proof of the analog fact for hilbert spaces ...
2
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2answers
36 views

Sequence of bounded linear operators implicating Cauchy sequence in $\mathbb K$

Let H be a Hilbert space and $(T_n)_{n \in \mathbb N}$ be a sequence in ${\rm BL}(H)$ (bounded linear operators) such that $(\langle y,T_nx \rangle)_{n \in \mathbb N}$ is a Cauchy sequence in $\mathbb ...
2
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1answer
78 views

Dual of $div$ on spaces where the tangential value is fixed

Say $\Omega$ is a domain in $\mathbb R^3$ with a smooth boundary $\Gamma$. Consider the spaces $$ H_{n,0}=\{v\in H^1(\Omega):n\cdot v \bigr |_{\Gamma} = 0\} $$ and $$ H_{t,0}=\{v\in ...
2
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1answer
45 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 ...
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 ...
2
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1answer
86 views

Conditional expectation as an orthogonal projection to what subspace?

Given a random variable $X$ and a sub sigma algebra $N$ of its sampling space, it is often said that $E(\dot \, \mid N)$ is an orthogonal projection, since $X-E(X\mid N)$ and $E( X\mid N)$ are ...
2
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1answer
37 views

Unbounded extension of bounded operator

Is it possible to construct an unbounded extension of bounded densely defined operator? To be more concrete, let $\mathcal{H}$ be Hilbert space, $\mathcal{D}\subset\mathcal{H}$ - a dense subset, ...
2
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1answer
84 views

How to find adjoint operator?

Let $(X,\langle\cdot,\cdot\rangle)$ be a Hilbert Space over $K$ with orthonormal basis $(x_n)$, and let $(\lambda_n)\in K$ be a bounded sequence. The mapping $T:X\to X$ is defined by ...
2
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2answers
89 views

Is $H^2\cap H_0^1$ equipped with the norm $\|f'\|_{L^2}$ complete?

Let $-\infty<a<b<+\infty$. Consider the norms $\|\cdot\|_{L^2}$, $\|\cdot\|_0$ and $\|\cdot\|_{H^1}$ defined on suitable spaces and given by ...
2
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2answers
113 views

Linear operators with no adjoint

Here is a standard theorem about bounded operators: Let $H$ be a Hilbert space. For any bounded linear operator $A:H\to H$ there is a unique bounded operator $A^*$ s.t $\langle Au,v\rangle=\langle ...
2
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2answers
71 views

Clarifying the definition of essential self-adjointness

If a Hilbert space operator $T$ has a unique self-adjoint extension, must the extension be the closure of $T$?
2
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1answer
61 views

determinant identity for invertible finite rank operators

I am currently reading a paper where the following identity, valid for an invertible finite - rank operator $T \colon \mathscr{H} \to \mathscr{H}$ on a separable Hilbert space, is given: $$ \log \det ...
2
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1answer
470 views

Orthonormal Basis for Hilbert Spaces

The following is the definition of orthonormal base that I am using: The notion of an orthonormal basis from linear algebra generalizes over to the case of Hilbert spaces. In a Hilbert space H, an ...
2
votes
2answers
240 views

Is fractional Sobolev space $H^s$ Hilbert?

For $s \in (0,\infty)$ a fractional number, define $H^s(\Omega) = W^{s,2}(\Omega)$ on good domain $\Omega$. Every textbook doesn't say that $H^s$ is Hilbert. Is it? I have only seen this fact when ...
2
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1answer
69 views

Closed range in Hilbert Space

If $H$ is a Hilbert Space. Let $A: H \rightarrow H$ be a one-to-one bounded operator with the additional property that $\beta||u|| \leq ||Au||$. How would you show that $R(A)$ (the range of A) is ...
2
votes
1answer
58 views

Norm of the dual of the Tensor product of Hilbert spaces

Let $V$ and $W$ be Hilbert spaces, we can define inner product and induced norm on Tensor product of these spaces as: Let $v_1,v_2 \in V$,and $w_1,w_2 \in W$. then $(v_1 \otimes w_1, v_2 \otimes ...
2
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1answer
97 views

If E is a Hilbert space and $T \in B(E)$ is compact, show $T(E)$ does not contain a closed infinite dimensional subspace

It's the problem from "Essential Results of Functional Analysis," R.J. Zimmer, Chapter 3, problem 3.1. I try to prove this problem and I am confused with the condition "closed infinite dimensional." ...
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1answer
73 views

Question about SOT and compact operators

I need some help with functional analysis / Hilbert space theory. If you have a favorite text to recommend, please let me know~ Here is my question: Given $v_t$ be the "squeeze operator" on ...
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1answer
54 views

Corollary to Putnam's theorem

Suppose $T_1$ and $T_2$ are normal operators on Hilbert spaces $\mathcal H_1$ and $\mathcal H_2$, respectively. Putnam showed that if $X$ is an operator satisfying $T_2X=XT_1$, then $T_2^*X=XT_1^*$. ...
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1answer
57 views

Does projection onto a finite dimensional subspace commute with intersection of a decreasing sequence of subspaces: $\cap_i P_W(V_i)=P_W(\cap_i V_i)$?

Let $V$ to be an infinite dimensional linear space over some field $k$. (you can take $k=\mathbb{C}$, or further assume $V$ is a complex Hilbert space). And assume $W$ is a finite dimensional ...
2
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2answers
95 views

Find the fallacy in using the Cauchy–Schwarz inequality

Let $\int_{a}^{b}\frac{f(x)}{x}dx=k$, wherein $f(x),a,b,k$ are positive. According to the Cauchy–Schwarz inequality: $\int_{a}^{b}xf(x)dx=\int_{a}^{b}x^{2}\frac{f(x)}{x}dx\leq \left ( ...
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1answer
84 views

Question about Hilbert Schmidt theory

Let $V \subset H \subset V^*$ be a Gelfand triple with all spaces being Hilbert and separable. Suppose $A:V \to V^*$ is such that $$\langle Au,u \rangle_{V^*,V} \geq C\lVert u \rVert^2_{V}$$ and $A$ ...
2
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1answer
58 views

Is $L^2(0,T;H_n)$ compactly embedded in $L^2(0,T;H)$?

Let $H$ be a separable Hilbert space with basis $h_i.$ Let $$H_n := \text{span}\{h_1,...,h_n\}.$$ Questions: 1) Is $L^2(0,T;H_n)$ compactly embedded in $L^2(0,T;H)$? 2) Is $L^2(0,T;H_n^*)$ ...
2
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1answer
83 views

Question about bases in Hilbert spaces and subspaces

Let $H$ be a Hilbert space. Suppose I have a basis for $H$ called $\{h_j\}$. Define $$H_n := \text{span}\{h_1,...,h_n\}.$$ Suppose now I am given an orthonormal basis for $H$ called $\{v_j\}$. My ...
2
votes
1answer
65 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 ...
2
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1answer
168 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 $a(t) = \frac{1}{\sqrt[4]t}$ ...
2
votes
1answer
141 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 $x \in l^2$. ...