Articles of functional analysis

On Fredholm operator on Hilbert spaces

Let $u: H \to H’$ be a continuous linear operator and $H,H’$ be Hilbert spaces. Let $u^\ast$ denotes its adjoint. By definition, an operator $u$ is called Fredholm if and only if $\ker u$ has finite dimension and $\mathrm{im}(u)$ has finite codimension. My book states that “…$u$ is Fredholm if and only if $u(H)$ is […]

Banach-Space-Valued Analytic Functions

This is Chapter VII, $\S$3, exercise 4, from Conway’s book: A Course in Functional Analysis: Let $X$ be a Banach space and $G\subset \mathbb{C}$ an open subset. We say that $f: G \to X$ is analytic if the limit $$\lim_{h \to 0} \frac{ f(z+h)-f(z)}{h}$$ exist in $X$ for all $z \in G$. Prove that if […]

Continuous functions with values in separable Banach space dense in $L^{2}$?

Let $[0,1] \subset \mathbb R$ be a the compact interval in the real numbers $\mathbb R$. We know that $C([0,1] \to \mathbb R)$ (the continuous function on $[0,1]$ with values in $\mathbb R$) are dense in $L^{2} ([0,1] \to \mathbb R)$ (the usual Lebesgue space). Now consider the Lebesgue space of functions on $[0,1]$, that […]

Supremum of absolute value of the Fourier transform equals $1$, and it is attained exactly at $0$

Suppose that $f \in L^1(\mathbb{R}^n)$, $f \ge 0$, $\|f\|_{L^1} = 1$. How do I see that $\sup_{\xi\in\mathbb{R}^n} |\mathcal{F}(f)(\xi)| = 1$, and it is attained exactly at $0$?

Inequality on a general convex normed space

Assume $(X,\|\cdot\|)$ is a normed space with the following property: if $x \neq y \in X$ have norm 1 then $\|\frac{x+y}{2}\|<1$. (We then say that $X$ is strictly convex) Prove that if $C$ is a convex (though not necessarily close) subset of $X$ and $x_0 \notin C$ and put $r=d(x_0,C)$ then $\{y\in X|\|y-x_0\|\leq r\}\cap C$ […]

weak derivative of a nondifferentiable function

I am reading a book on Sobolev and having trouble understanding a notion of weak derivative. I consider a function $(x-1)^+=\max(x-1,0),x\in[0,2]$, I have a problem at $x=1$, so it is continuous and “almost everywhere differentiable” now, does it have a weak derivative everywhere? Can we “quantify” that? I know that the derivative in classical sense […]

Simplicity and isolation of the first eigenvalue associated with some differential operators

Consider the operator $\Delta$ or more generally a second order differential operator $L$, in which the principal part is symmetric and positive definite. It can be proved (see here page 336) that the first eigenvalue “$\lambda_1$” of $L$ is simple, i.e. the space generated by the eigenfunctions associated with $\lambda_1$ is one-dimensional. Moreover, $\lambda_1$ is […]

Prove: For any sequence of linearly independent elements $y_j \in X$ and $a_j \in \mathbb R$ there exists an element $f \in X^*$ s.t. $f(y_j)=a_j$

I’m trying to solve the following problem but I have no clue how to do it. Let $(X,||.||)$ be a normed $\mathbb C$-vector space. Prove: For any sequence of linearly independent elements $y_j, 1 \leq j \leq N$, in $X$ and any sequence $(a_j)_{1 \leq j \leq N}$ in $\mathbb R$ there exists an element […]

What kind of “isomorphisms” is the mapping from $H_0^1$ to $H^{-1}$ defined by the elliptic operator?

The following is an excerpt from Evans’s Partial Differential Equations Here $L$ is the elliptic operator. The part about $\langle f,v\rangle$ is stated in this question. Would anybody clarify what kind of “isomorphism” it should be in the last sentence? Of course it is a vector space isomorphism which means $L_\mu$ is a bijective linear […]

Compactness of Sobolev Space in L infinity

I want to show that if $u_{m} \rightharpoonup u$ in $W^{1,\infty}(\Omega)$ then $u_{m} \rightarrow u$ in $L^{\infty}(\Omega)$. I know that I can’t directly use the compactness of Rellich Kondrachov Theorem since I am taking $p = \infty$. From Morrey’s Inequality I have $||u||_{C^{0,\alpha}} \leq ||u||_{W^{1,p}}$ where $\alpha = 1 – \frac{n}{p}$. Is it possible to […]