Articles of functional analysis

Prove that $R: W^{\perp} \to (V/W)^*, \ Rx^* = \tilde{x}^*$ is a well-defined.

Let $(V, \| \cdot \|)$ be a normed vector space and $M^{\perp}$ be the annihilator of $M$. If $W\subset V$ is a closed linear subspace. Prove that $R: W^{\perp} \to (V/W)^*, \ Rx^* = \tilde{x}^*$, where $\tilde{x}^*(x+ W) = x^*(x)$ is well defined. I want to prove that when we apply $R$ on an element […]

How to reduce this to Sturm-Liouville form?

I have the ODE $$-(1-x^2) \frac{d^2 f(x)}{dx^2} + x \frac{df(x)}{dx}+g(x)f(x)=\lambda f(x)$$ and I want to reduce it to Sturm-liouville form. The problem is that we don’t have $2x$ but just $x$. otherwise it would be similar to the Legendre differential equation. Could anybody help me with that? By the way, does this mean that the […]

Characterization of compactness in weak* topology

Let $ X $ be Banach space, and $X^*$ its dual. A set $ F \subset X ^ * $ is weakly-* compact if and only if $ F $ is closed in the weak* topology and is bounded in norm. How does one prove this (well-known) fact? Notes This characterization of compactness in weak* […]

Compact subspace of a Banach space .

The following statement doesn’t make sense to me, can someone justify it to me ? If $K$ is a compact subset of a Banach space $Y$ then there exists for $\epsilon > 0 $ a finite dimensional subspace $Y’$ of $Y$ such that $d(x, Y’) < \epsilon $ for every $x \in K $ , […]

When Schrodinger operator has discrete spectrum?

On $L^{2}(\mathbb{R})$ we have a linear operator $S=-\frac{d^{2}}{dx^{2}}+u(x)$. As I understand for some choices of potential $u$ (like harmonic oscillator $u(x)=\frac{\omega x^2}{2}$) Schrodinger operator will have only a countable set of $\lambda_{n} \in \mathbb{R}, f_{n} \in L^{2}(\mathbb{R}), (f_{n} \neq 0)$ such that $S f_{n}=\lambda_{n} f_{n} \\$. My question is for what other choices (if any) […]

Two (equivalent ?) norms on Hilbert space

Let $H$ be a vector space equipped with two inner products $\langle \cdot,\cdot\rangle_1, \ \langle \cdot,\cdot\rangle_2$, $(H,\langle \cdot,\cdot\rangle_j)$ is a Hilbert space for $j=1,2$. Suppose that $R_1$ and $R_2$ are linear operators on $H$, which are bounded w.r.t. $\|\cdot\|_1$ and $\|\cdot\|_2$,respectively. Moreover, let us assume that $R_1$ is injective and for any $x,y\in H […]

Why sequential continuity from $E$ to $E'$ implies continuity?

Let $(E,\|\cdot\|)$ be a separable Banach space. Let $E’$ be the topological dual of $E$ equipped with the weak* topology $w^*$. I read that a certain linear operator $J:(E,\|\cdot\|)\to (E’,w^*)$ is continuous because $J(u_n)\xrightarrow{w^*}J(u)$ provided that $u_n\xrightarrow{\|\cdot\|}u$. So, the proof consists in proving that the operator is sequentially continuous. I’d like to justify that, in […]

Lower semi-continuous function which is unbounded on compact set.

Every lower semi-continuous functions attains an infimum/minimum on a compact set, do you know examples of lower semi-continuous functions which are unbounded and/or don’t attain their maximum/supremum?

Inequality in Mercer's theorem proof

In the outline of the Mercer’s theorem proof there is an inequality assumed without any explanation: $$\sum_{i=0}^{\infty} \lambda_i \vert e_i(t) e_i(s) \vert \le \sup_{x \in [a,b]} \vert K(x,x)\vert^2$$ Why does this need to hold?

Sums of projections in a C*-algebra

Let $A$ be a $C^*$-algebra, and let $p_1, \ldots, p_n \in A$ be projections, meaning $p_i = p_i^* = p_i^2$. Now assume that the sum $p = p_1 + \ldots + p_n$ is also a projection. How can one show that this implies that the $p_i$’s must be orthogonal, i.e. $p_ip_j = 0$ whenever $i […]