Articles of functional analysis

Approximation by smooth function while preserving the zero set

Let $\mathbb{T}$ denote the unit circle. Given $f \in \mathcal{C}(\mathbb{T})$, can we approximate $f$ by smooth functions having the same zero set ? i.e. for $\varepsilon >0$, can we find $g \in \mathcal{C}^\infty(\mathbb{T})$ such that f(x) = 0 if and only if $g(x) = 0$; $\|f-g\|_\infty < \varepsilon$. Both tasks can easily be performed separately. […]

Function invariant under Hilbert transform

Let $f\in C_0^\infty(\mathbb{R}^n)$ and $$Hf(x)= \operatorname{p.v.}\int_{\mathbb{R}}\frac{f(x-y)}{y} \, dx$$ the Hilbert transform of $f$. Is it possible that $Hf=f$ (a.e. and possibly after extending $H$ to $L^p$ space) ?

Finding the derivative of the norm

Consider function from Hilbert space to real numbers. $F(x)=\| Ax\|$. My question how to find it’s derivative $F'(x)$.

How common is it for a densely-defined linear functional to be closed?

I’ve always held the vague belief that any densely-defined operator encountered “in nature”, if it isn’t bounded, is probably at least closable. But, today I noticed the following thing: Consider the Banach space $C_0(\mathbb{R})$ of continuous, complex-valued function vanishing at $\pm \infty$ in the uniform norm. We have a densely-defined linear functional $\int : C_c(\mathbb{R}) […]

Norm of the linear functional

Could you help me, please with the following question? There is a linear functional $A : C_{[0;1]} \rightarrow \mathbb{R}$, such that $$ Ax=\int_{a}^{b}x(t)\varphi(t)dt $$ where $\varphi$ is a fixed fucntion, $\varphi \in C_{[0;1]}$. The task is to prove that $ \left \| A \right \| = \int_{a}^{b}\left | \varphi (t) \right |dt $. The norm […]

Fourier Transform: Understanding change of basis property with ideas from linear algebra

The notion of Fourier transform was always a little bit mysterious to me and recently I was introduced to functional analysis. I am a beginner in this field but still I am almost seeing that the Fourier transform can be viewed as a change of basis in a space of functions. I read the following […]

A Banach space is reflexive if a closed subspace and its quotient space are both reflexive

Let $X$ be a Banach space. Let $Y$ be a closed subspace. Suppose that the normed spaces (in fact Banach spaces) $Y$ and $X/Y$ are both reflexive. I need to show that $X$ is reflexive. I cannot show this but I feel that I could use the fact that a Banach space is reflexive if […]

Exercise books in functional analysis

I’m studying functional analysis and I was wondering if there are some exercise books (that is, books with solved problems and exercises) The books I’m searching for should be: full of hard, non-obvious, non-common, and thought-provoking problems; rich of complete, step by step, rigorous, and enlightening solutions;

Ref. Requst: Space of bounded Lipschitz functions is separable if the domain is separable.

I have been scouring the internet for answers for some time and would therefore appreciate a reference or a proof since i’m not able to produce one myself. Let $(\mathcal{X},d)$ be a metric space, and let $$ BL(\mathcal{X})=\{f:\mathcal{X}\to \mathbb{R}\, \, | \, \, f \text{ is Lipschitz and bounded}\} $$ denote the bounded real-valued Lipschitz […]

Why is the numerical range of a self-adjoint operator an interval?

I was reviewing for a test for functional analysis when I came across the following statement: Let $T$ be a bounded self-adjoint operator on a Hilbert space $H$. Then the numerical range of it is an interval $[m, M]$ with $M>0$. Is the above statement correct? How can I prove it? Thank you!!