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

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Dense subspace of $L^{2}[0,1]$

I know that $C[0,1]$ is dense in $L^{2}[0,1]$ but is $\{f\in C^{2}[0,1]:f(0)=f(1)=0\}$ dense in $L^{2}[0,1]$?
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1k views

Example that in a normed space, weak convergence does not implies strong convergence.

The book "Introductory Functional Analysis with Applications" (Kreyszig) presents the following definitions. A sequence $(x_n)$ in a normed space $X$ is said to be strongly convergent if there is ...
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261 views

Counterexample for the stability of orthogonal projections

Let $V$ be a seperable Banach space, which is dense and continuously embedded in a Hilbert Space $H$. Let $(V_m)$ be a Galerkin scheme (See definition below) for $V$. Using the embedding we can ...
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1answer
614 views

A few questions about the Hilbert triple/Gelfand triple

I am attempting to fully understand Hilbert triples by reading Brezis' Function Analysis book. Consider $V \subset H \subset V^*$, where $V$ is Banach and $H$ is Hilbert. $V$ is dense in $H$. ...
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320 views

For what sequences of real numbers $\left\{ k_{n}\right\}$ is the set of functions $\left\{ e^{ik_{n}x}\right\}$ a basis?

It is well known that the set of functions $\left\{ e^{^{inx}}\right\}$, for integer $n$, is an othonormal basis for the space of square integrable real functions in the interval $[-\pi,\pi]$. Now ...
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120 views

Is $\mathcal{C}([0,1])$ homeomorphic to a Hilbert space?

Let $\mathcal{C}([0,1])$ the Banach space of continuous functions from $[0,1]$ to $\mathbb{C}$. The norm on $\mathcal{C}([0,1])$ is $f \mapsto \| f\|_{\infty}= \sup_{x \in [0,1]} |f(x)|$. Is it ...
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837 views

Contexts For Bessel's Inequality?

Bessel's inequality appears to be about orthonormal sequences. But (in the context of inner product spaces), I've thought of this inequality as being a demonstration that the hypotenuse of triangles ...
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100 views

Tight Probability on Hilbert space

I am considering the following problem. Let $(X_j)$ be i.i.d. $N(0,1)$ random variables and $H$ a Hilbert space with orthonormal basis $(e_j)$. Let $$X:=\sum_j \frac{X_j e_j}{j}$$ And for any ...
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214 views

Reproducing Kernel Hilbert Space is dense?

Let $E=C[0,1]$, space of all real-valued continuous functions on $[0,1]$, $\mathcal{E}$ be its Borel $\sigma$-algebra and $\mu$ a Gaussian measure on $E$. Let $E^*$ be a space of all continuous ...
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402 views

On the weak and strong convergence of an iterative sequence

I have some difficulties in the following problem. I would like to thank for all kind help and construction. Let $H$ be an infinite dimensional real Hilbert space and $F: H\rightarrow H$ be a ...
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322 views

A complete orthonormal system contained in a dense sub-space.

Let H be a separable complex Hilbert space. Let A be a dense sub-space of H. Is it possible to find a complete orthonormal system for H that is contained in A?
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322 views

Question about positive operators on a Hilbert space

I have the following problem. Let $\Omega \subset R^n$ have finite measure, let $H = L^2(\Omega)$ and let $S: H \to H$ be a bounded linear operator. Then it is well known that $P = SS^*$ is a positive ...
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133 views

If a linear operator has an adjoint operator, it is bounded

This is a question I'm struggling with for a while: Let $H$ be a Hilber space. Let $T,S: H\rightarrow H$ be linear operators (not neccessarily bounded) such that for every $x,y\in H$: $\langle ...
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1answer
1k views

Compactness of Multiplication Operator on $L^2$

Suppose we have an bounded linear operator A that operates from $L^2([a,b]) \mapsto L^2([a,b])$. Now suppose that $A(f)(t) = tf(t)$. Is A compact? Edit: I know $A = A^*$ but I'm not really sure ...
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359 views

Weak Formulations and Lax Milgram:

I have a question on how to put a PDE into weak form, and more importantly, how to properly choose the space of test functions. I know that for an elliptic problem, we want to start with a problem ...
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323 views

Why isn't it a Hilbert space

Let $X$ be the vector space of all the continuous complex-valued functions on $[0,1]$. Then $X$ has an inner product $$(f,g) = \int_0^1 f(t)\overline{g(t)} dt$$ to make it an inner product space. But ...
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257 views

Example of a non-algebraic $\ell^2$-function in two variables

Let's call an $\ell^2$-function $\mathbb{N} \times \mathbb{N} \to \mathbb{C}$ algebraic if it is in the image of the natural algebra homomorphism $\ell^2(\mathbb{N}) \otimes \ell^2(\mathbb{N}) \to ...
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136 views

Non-examples for the Kato-Rellich Theorem

The Kato-Rellich Theorem is a classical result stating that if $A,B$ are unbounded operators on a Hilbert space with $A$ self-adjoint, $B$ symmetric, $\mathcal D (A)\subset \mathcal D(B)$ and $$ ...
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48 views

Derive Fourier transform from what it should do?

I was wondering about the following: Imagine you want to figure out whether there is a transform that exchanges differentiation with multiplication and convoution with pairwise transformation for ...
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1answer
247 views

Approximating a Hilbert-Schmidt operator

Let $H$ be a separable Hilbert space. Recall that a bounded operator $A : H \to H$ is said to be Hilbert-Schmidt if $$\|A\|_{HS}^2 := \sum_{i=1}^\infty \|A e_i\|^2 < \infty$$ where ...
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423 views

On the isometry between bounded linear operators and the dual of nuclear linear operators

Let $H$ be a separable Hilbert space. Let $(e_i)_i$ be an orthonormal basis. For any bounded linear map $T$ we write, whenever possible $$\operatorname{tr} T := \sum_{i}^{\infty} \langle T e_i, e_i ...
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208 views

Measure on a separable Hilbert space

Let $H$ be a real separable Hilbert space. Is it true that there exist a probability space $(\Omega, \mu)$ and a measurable function $\pi\colon \Omega \to H$ such that for any $h \in H$ we have $$ ...
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140 views

A complete eigenvector basis for the restricted operator

Let $X$ be a (not necessarily bounded) selfadjoint linear operator on a Hilbert space $H$ and let $M$ be a closed subspace such that $X(M) \subset M$. Suppose that $X$ admits an orthonormal basis ...
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266 views

Two different definitions of ellipticity

This is a question originating in another mathematics forum, matematicamente.it (in Italian). In literature one encounters the word elliptic in (at least) two different definitions. In what follows ...
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73 views

Show that an unbouned normal operator is closed

A linear operator $A$ is called nomal if $\mathcal{D}(A)=\mathcal{D}(A^{*})$ and $\lVert A\phi\rVert =\lVert A^{*}\phi\rVert$ for every $\phi\in \mathcal{D}(A)$. Show that normal operators are closed. ...
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How are Hilbert Space methods used in number theory?

In N. Young's book "An introduction to Hilbert Space," there is an interlude in which the author remarks that the theory of Hilbert spaces is "routinely used in differential geometry, complex ...
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292 views

Is this question solvable? $2$ non-linear equations and the proof that the solution is unique (with asymmetric bounty option)

As mentioned in the title I want to show the uniqueness of the solution to $2$ non-linear equations. However, it seems that I can not solve this question with my current mathematical knowledge. More ...
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374 views

Why is the numerical range of an operator convex?

Let $T$ be a Hilbert space operator. Its numerical range is \begin{equation} W(T)=\{\langle Tx,x\rangle:\|x\|=1\}.\end{equation} It is a well-known fact that $W(T)$ is a convex subset of the complex ...
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1k views

Elegant proof that $L^2([a,b])$ is separable

Is anybody aware of, or can provide at least an outline, of a proof that the Hilbert space of Lebesgue functions square-integrable on the closed real interval [a,b], equipped with the $L^2$ norm, is ...
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932 views

Relationship of Fourier series and Hilbert spaces?

I just read in a textbook that a Hilbert space can be defined or represented by an appropriate Fourier series. How might that be? Is it because a Fourier series is an infinite series that adequately ...
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180 views

Hilbert space with all subspaces closed

Does there exist an infinite-dimensional Hilbert space with all subspaces closed?
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397 views

Soft Question Hilbert Space Geometry

Just a quick question about the geometry of Hilbert spaces from an intuitive standpoint. Maybe just assuming we're working with $L^2$ would simplify the situation. Basically, in something like ...
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206 views

Sequences are Cauchy depending on the norm?

Assume that we have two Banach spaces $B_1, B_2$ with their underlying sets being identical. Is it possible that a Cauchy sequence in one of the spaces would fail to be Cauchy in the other? I am ...
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496 views

$f$ an isometry from a hilbert space $H$ to itself such that $f(0)=0$ then $f$ linear.

This question was on an exam and I am not sure how to answer it. I mostly tried writing zero in different ways and tried lots of algebra to get something out. I also tried to use the fact that $H$ is ...
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1answer
180 views

Can the composite of two projections really fail to be a projection?

Let $H$ denote a Hilbert space. For any closed subspace $C \subseteq H$, write $P_C$ for the orthogonal projection onto $C$. Then according to wikipedia, the composite $P_U \circ P_V$ needn't be a ...
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502 views

Norm inequality for sum and difference of positive-definite matrices

If $X_{1}$ and $X_{2}$ are positive definite matrices, how to show that $\left\Vert X_{1}-X_{2}\right\Vert \le\left\Vert X_{1}+X_{2}\right\Vert$ for the spectral norm? and how about for the nuclear ...
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192 views

Homeomorphisms between infinite-dimensional Banach spaces and their spheres

As I know Cz. Bessaga has proved that an infinite-dimensional Banach space is homeomorphic to its unit sphere. Unfortunately I do not have his book but I want to know is this theorem true without ...
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193 views

Show that linear Operator on $\ell^2$ is unbounded

Currently, I am preparing for a next semester course and trying to figure out some basic concepts in functional analysis. Let $T:\mathcal{D}(T)\to \ell^2$ be defined by ...
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1answer
314 views

Uniform mean ergodic theorem

I'm working in Einsiedler and Ward's book on Ergodic Theory and in Exercise 2.5.4 they want to prove the following $$\lim_{N - M \to \infty} \frac{1}{N - M} \sum_{n = M}^{N - 1} U_T^n f \to P_T f.$$ ...
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96 views

What's the spectrum of this operator in $\ell^2$?

Suppose that $\ell^2 = \biggl\{(x_n)_n \in \mathbb{K}^{\mathbb{N}_0} \biggm| \sum_{n=1}^{\infty}|{x_n}^2| < +\infty \biggr\}$ is a Hilbert-space with the inproduct $\langle\cdot,\cdot\rangle_2: ...
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114 views

Norm of a $2\times 2$ matrix as a Hilbert space operator

Work in the Hilbert space $\mathbb C^2$. Let $$A = \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix} $$ be a matrix with entries in $\mathbb C$, and let $A$ ...
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300 views

Unit Ball of $\mathcal{l}_2$

Let $B(\mathcal{l}_2) :=\{x \in \mathcal{l}_2 : \|x \| \leq 1 \}$ and $S(\mathcal{l}_2) :=\{x \in \mathcal{l}_2 : \|x \| = 1 \}$ be the unit ball and the unit sphere of $\mathcal{l}_2$, respectively. ...
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431 views

Reproducing kernel Hilbert spaces and the isomorphism theorem

A reproducing kernel Hilbert space is a Hilbert space in which the evaluation functional $L_x : f \rightarrow f(x)$ is continuous. By continuity, the Riesz representation theorem says that this ...
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74 views

Show that $\|e^{tA}\| \le e^{t\|\Re (A)\|}$

Let $X$ be a complex Hilbert space, and let $A$ be a bounded linear operator on $X$. Define the real part of $A$ to be $\Re(A)=\frac{1}{2}(A^{\star}+A)$, and define ...
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756 views

Tensor product of operators

We know that if $T_1$ is a linear bounded operator on a Hilbert space $H_1$ and $T_2$ is a linear bounded operator on a Hilbert space $H_2$ there exists a unique linear bounded operator $T$ on $H_1 ...
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660 views

Parallelogram law valid in banach spaces?

It is known that the parallelogram law $\|x-y\|^2+\|x+y\|^2 = 2(\|x\|^2 + \|y\|^2)$ holds in any space with an inner product (the norm being induced by this inner product). Is this formula valid in ...
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149 views

Convex subset of Hilbert space as intersection of closed balls

How does one prove that any closed, convex, and bounded subset of a Hilbert space is the intersection of the closed balls that contain it?
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1answer
510 views

Exhibiting open covers with no finite subcovers.

How do I exhibit an open cover of the closed unit ball of the following: (a) $X = \ell^2$ (b) $X=C[0,1]$ (c) $X= L^2[0,1]$ that has no finite subcover?
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593 views

Recognition of an orthonormal complete set in an Hilbert Space

I recently came across this exercise: Let $u_n:[0,1]\to \mathbb R$ be the sequence of functions defined by: $$(u_n):=\text{sign }(\sin(2^n\pi x)),\qquad n=0,1,2,\dots$$ a)Prove that this set of ...
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391 views

Sum of Closed Operators Closable?

Let $A$ and $B$ be closed operators on a (separable complex) Hilbert space with dense domains $D(A)$ and $D(B)$ respecitvely. Then, we may define the operator $A+B$ on $D(A)\cap D(B)$. In general, ...