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

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A function $w(t)$ which satisfy $\int dt w(t)F[x](t)=c$

Consider a differentiable scalar function in two variables $F(x,t)$ for $x\in X$ and $t\in T$, then it can be viewed as an infinite family of scalar functions $\{F[x](t))\}_{x\in X}$. Are there any ...
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1answer
23 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 ...
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1answer
22 views

How to show: $\exists$ a subspace $L$ of a Hilbert space $H$ such that $H\neq L\oplus L^{\perp}$.

How to show: $\exists$ a subspace $L$ of a Hilbert space $H$ such that $H\neq L\oplus L^{\perp}$. Let consider $H=l_2$ where $l_2=\lbrace x=(x_n)^\infty_1: \sum^\infty_1 |x_n|^2<\infty \rbrace $ ...
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2answers
17 views

Orthonormal zero Function

I have this exercise Let H be a Hilbert space with orthonormal basis $\{e_n | n\in N\}$ and let $f_n = e_n + e_{n+1}$ If $\langle f,f_n \rangle = 0$ for all $n$ how do I prove that $f=0$ I think i ...
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1answer
30 views

how to show this function doesn't belong to Hilbert space?

I am trying to show $\chi_{B_R(0)}(x) \notin H^1 (\mathbb{R}^n)$ , ∀R>0. since $H^1 (\mathbb{R}^n) := W^{1,2}(\mathbb{R}^n)$ That is, I have to show that $\chi_{B_R(0)} (x) \notin ...
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1answer
22 views

Trace norm of Hermitian matrix

Let $A\in L(H)$ some Hermitian matrix, where $H$ is some finite dimensional Hilbertspace. I want to show $$\left\|A\right\|_{tr} = \max_{U\in U(H)}|\text{tr}(UA)| \ \ \ (*)$$ where U is unitary, and ...
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2answers
41 views

Determining if the span of a set is dense in L^2(0,1)

I am trying to determine whether or not the following statement is true: If $f \in L^2(0,1)$ and $\int_0^1 x^nf(x) = 0$ for all positive integers $n$. Then $f(x) = 0$ I have already verified this ...
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0answers
31 views

Bounded linear functionals on $L^\infty$.

I am looking at a practice final and I am a bit confused by this statement I am trying to prove: "There is a nonzero bounded linear functional on $L^\infty[0,1]$ which vanishes on the subspace ...
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1answer
33 views

Applying perp twice in a hilbert space

Let $H$ be a hilbert space and let $K \subset H$ be a subspace. Then $\overline{K} \subset K^{\perp\perp}$, but why does the reverse inclusion hold?
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1answer
22 views

Show compactness of an operator with Arzelà–Ascoli

We have $K\colon L^{2}(a,b) \rightarrow L^{2}(a,b)$ such that $ Kf(t)=\sum_{j=1}^{n}\phi_{j}(t) \int_{a}^{b} \psi_{j}(S) f(s)ds$ where $\phi_{j} ,\psi_{j} \in L^{2}(a,b)$. We want to show that K is ...
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1answer
17 views

Nonexpansive Affine Operators in Hilbert spaces

Let $H$ be a Hilbert space with real inner product $\langle \cdot, \cdot \rangle : H \times H \rightarrow \mathbb{R}$. Let $T$ be an affine operator. Show that $T$ is nonexpansive, i.e., $\left\| ...
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1answer
34 views

Invertible iff Bounded below and dense range

Statement: Given a Hilbert space $\mathscr{H}$ and $\mathscr{K}$ and a bounded operator $A \in \mathscr{B}(\mathscr{H}, \mathscr{K})$. Show that $A$ is invertible if and only if $A$ is bounded below ...
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20 views

Convex Feasibility problem on Hilbert space

Let $H$ be a Hilbert space with real inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$. Consider a closed convex set $X \subset H$ and let $P_X: H \rightarrow X$ be the ...
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55 views
+50

Projection and Pseudocontraction on Hilbert space

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let ...
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1answer
31 views

Hilbert space inequality $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$

In prelim prep I came across 'given $\epsilon$ there exists $C_{\epsilon}$ such that $|\langle x,y\rangle | \leq \epsilon ||x||^2 + C_{\epsilon}||y||^2$. It is asserted without proof, so I've tried ...
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20 views

Does Hilbert space with countable dimensions exist? [duplicate]

If there is a Hilbert space with infinite dimensions, can it have countably infinite dimensions? And does Banach space with countable dimensions exist?
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20 views

Limit of exp of self-adjoint operator

Let $A$ be self-adjoint (possibly unbounded) operator on Hilbert space $\mathcal{H}$. Under what conditions $w-\lim_{t\rightarrow\infty} e^{i A t}=P_0$, where $w-\lim$ - the limit in weak operator ...
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0answers
11 views

Hilbert-Schmidt theorem

In the Hilbert-Schmidt theorem what it means : $A e_n=\lambda_n e_n$ ? Thank you .
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1answer
37 views

Need help proving the equivalence of two norms !

Hey I could use alot of help with this problem please! Let (X, <-,->) be a Hilbert space over R. Then, let A: X -> X be a linear operator. Suppose that A is symettric and positive definite. Let ...
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2answers
57 views

Isometry <=> Adjoint left inverse [duplicate]

Is it true that: $$T\text{ isometric}\iff T^*\text{ left inverse}$$ Obviously: $$\text{"}\Rightarrow\text{": }\langle x,\tilde{x}\rangle=\langle Tx,T\tilde{x}\rangle=\langle x,T^*T\tilde{x}\rangle$$ ...
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2answers
44 views

Isometry: Adjoint = Leftinverse

Is it true that: $$T\text{ isometric}\iff T^*\text{ left inverse}$$ Obviously: $$\text{"}\Rightarrow\text{": }\langle x,\tilde{x}\rangle=\langle Tx,T\tilde{x}\rangle=\langle x,T^*T\tilde{x}\rangle$$ ...
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1answer
17 views

Is $\phi(h)=\displaystyle \sup_{x\in X}{\|h-x\|}$ continuous on Hilbert space when $X$ is bounded?

Let $H$ be a hilbert space, and let $X \subset H$ be a bounded subset of $H$. Let define the function $\phi:H \to H$ by the rule $\phi(h)=\displaystyle \sup_{x\in X}{\|h-x\|}$. I want to know if this ...
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1answer
24 views

I want to show one norm is less than or equal to another norm on C([0,1])

Let $|| \ ||_1$ be the norm on $C([0,1])$ defined by $||f||_1 = \int_0^1|f(t)|dt$. a) Show that $||f||_1 \le ||f||_{[0,1]}$ b) Are $|| \ ||_1$ and $|| \ ||_{[0,1]}$ equivalent? For part a) I think ...
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1answer
31 views

Alternating projections on a Hilbert space

Let $P_1, P_2$ be the orthoprojections onto $S_1, S_2$, closed subspaces of a Hilbert space $H$. It is straightforward to show that if $(P_1P_2)^nx \to z$ then $z \in S_1 \cap S_2$ (I can post a quick ...
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1answer
22 views

If $x_n \to x$ in $L^1(X,H)$ then $\langle x_n, h \rangle \to \langle x, h \rangle$ in $L^1(X,\mathbb{R})$

Suppose $H$ is a Hilbert space. Is it true that if $x_n \to x$ in $L^1(X,H)$ then $\langle x_n, h \rangle \to \langle x, h \rangle$ in $L^1(X,\mathbb{R})$ for any fixed $h\in H$? Certainly if $x_n\to ...
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34 views

How do I integrate $\langle\nabla u,\nabla v \rangle$ in arbitrary dimensions?

I am trying to show that if $u_n$ are eigenfunctions of the Laplacian operator that make up an orthonormal basis of $L^2$, then $u_n\sqrt{\lambda_n}^{-1}$ form an orthonormal basis of $H^1_0$. I ...
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11 views

Functions in a Reproducing Kernel Hilbert Space are Lipschitz continuous

I would like to show that all the functions in a Reproducing Kernel Hilbert Space (RKHS) are Lipschitz continuous. So that, I take two points in the domain $\vec{x}_{1} ,\vec{x}_{2} \in X$ then from ...
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25 views

Finding the norm of a linear operation.

I am reading A course in real analysis by John McDonald, on page 530, it says "it is easy to show $|||J|||=1$" where $J$ is the linear operation $J:C([0,1])\rightarrow C([0,1])$, defined by $J(f)(x) = ...
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1answer
9 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
17 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
32 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
23 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
30 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 ...
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1answer
13 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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39 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
15 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
52 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
70 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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27 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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12 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
39 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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38 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
19 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
45 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~ ...
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1answer
30 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
31 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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34 views

Composition property of the functional calculus

Prove from the spectral theorem for normal operators $T$ that for bounded borel functions $f,g$ we have $f{\circ}g(T)$=$f(g(T))$.
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1answer
48 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
50 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
48 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 ...