Articles of functional analysis

Derivative of Rayleigh quotient

I’m going over the proof of the spectral theorem for compact symmetric operators in Hilbert space in Lax. Let $A$ be a compact symmetric operator on a Hilbert space to itself. Define the Rayleigh quotient to be $$R_A(x) = \frac{(Ax, x)}{\|x\|^2}$$ Let $z$ be vector that maximizes the quadratic form $(Ax, x)$ over the unit […]

Show that $ \Lambda:a\mapsto \Lambda_a $ defines a linear map from $ l^1(\mathbb{N}) $ to $ c^0(\mathbb{N})^* $

Given $ a=(a_n)_{n\in \mathbb{N}}\in l^1(\mathbb{N}) $, define for each $ c=(c_n)_{n\in \mathbb{N}}\in c^0(\mathbb{N}) $ $$ \Lambda_a(c)=\sum_{n=1}^{\infty}a_nc_n. $$ I want to show that $ \Lambda:a\mapsto \Lambda_a $ defines a linear map from $ l^1(\mathbb{N}) $ to $ c^0(\mathbb{N})^* $, where $ c^0(\mathbb{N})^* $ denotes the dual of $ c^0(\mathbb{N}) $ (see the definition in my answer). […]

$L^2$-convergence of a sequence of step functions of differences

Suppose $f\in L^2=L^2([0,1])$ (that is, $f$ is square-integrable). Let $f_n$ be an approximation of $f$ by steps of differences of $\int f$. Formally, $$f_n = \sum_{i=1}^n\mathbf{1}_{((i-1)/n,i/n]}\left(n\int_{(i-1)/n}^{i/n}f(t)dt\right)$$ Explanation: I have divided $[0,1]$ into $n$ equal parts, and let $f_n$ be constant on each part, with height equal to $\int f$’s difference on the edges of that […]

When the point spectrum is discrete?

Are there some criteria to tell when the point spectrum of a linear operator is discrete? In general it is not the same (take the spectrum of the “annihilation” operator). More specifically, what are the conditions that should be satisfied by a symmetric (or even self-adjoint) operator to have “point spectrum” = “discrete spectrum”?

Construct dense subspace of codimension $n$ for all $n$

I want to prove the following: Let $X$ be an infinite dimensional normed space. For all integer $n\geq1$: $X$ has a dense subspace of codimension $n$, i.e. a subspace $L$ such that $\dim(X/L)=n$. How can i do this? My first thought was: Take a basis $e_1,e_2,\ldots$ such that $X=Span(e_1,e_2,\ldots)$ and let $L=Span(e_2,e_3,\ldots)$ than $\dim(X/L)=1$. On […]

Fourier Transform calculation

I am trying to calculate the Fourier Transform of $$f(x)=\exp(-\frac{|x|^2}{2}). $$ Thus, I am looking at the integral $$ \hat{f}(u)=\int_{\mathbb{R}^n} \exp(-\frac{|x|^2}{2}) \cdot \exp(ix\cdot u) dx. $$ I can’t figure out how to evaluate this integral. Am I trying the wrong approach to calculate the transform or should I be able the integral. Note the integral […]

Nonconvex set converging to a convex set despite holes

I’m looking at the example in Figure 4-7 of “Variational Analysis” (Rockafellar and Wets). Basically, there’s a sequence of sets $C_{\nu}$ riddled with holes, and it states that the sequence eventually converges to the set $C$ (with the same shape but without holes) as long as the holes get finer and finer and thus vanish […]

Is this space complete?

Let $X$ be the space of measurable functions $f:[0,1] \rightarrow \mathbb{R}$. I want to find out whether this space is complete under the metric $d(f,g):= \int_{[0,1]} \frac{|f-g|}{1 + |f-g|}$. Does anybody here have an idea how to tackle this problem?

Universal $C^*$-algebra of countable family of self-adjoint operators have boundedly complete standard Schauder basis

Let $A = C^*(T_1,T_2,T_3,… | T_i=T_i^*, ||T_i|| \leqslant 1)$ – universal enveloped $C^*$-algebra of countable family of self-adjoint operators. $A$ have standard Schauder basis, which contains all words generated by $T_i$ (i.e. $T_1 T_3 T_2 T_5$ – is basis vector). Is it true, that it is boundedly complete Schauder basis or monotonically boundedly complete Schauder […]

Preservation of Lipschitz Constant by Convolutions

The following is a step in a proof: $f$ is a Lipschitz function from $E$ to $F$ where $E$ is a finite-dimensional Banach space and $F$ an arbitrary Banach space. $\phi\geq 0$ is a $C^\infty$ function with compact support, $\int\phi=1$, $\phi(x)=\phi(-x)$. The function $g(z)=\int f(z+x)\phi(x)dx$ is defined. The claim is that $||g||_{\text{Lip}}\leq||f||_{\text{Lip}}$. I see why […]