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Questions tagged [functional-analysis]

Functional analysis, the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces and other topics. For basic questions about functions use more suitable tags like (functions), (functional-equations) or (elementary-set-theory).

Functional analysis is the study of infinite-dimensional vector spaces, often with additional structures (inner product, norm, topology), with typical examples given by function spaces. The subject also includes the study of linear and non-linear operators on these spaces, including spectral theory, as well as measure, integration, probability on infinite dimensions, and also manifolds with local structure modeled by these vector spaces.

For basic questions about functions use more suitable tags like , or .

52582 questions
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A bounded sequence in metric space that has no convergent sequence

Let consider first the space $$V=\{(x_1,x_2,...)\mid \sup_i |x_i|<\infty \}.$$ This is a complete metric space (refer to my course). Now I was wondering if the set $$\mathcal A=\{x\in \ell^\infty \mid \|x\|_{\ell^\infty }\leq 1\},$$ is compact. I…
user380364
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Spectrum of the operator

Let $T$ be an operator on Hilbert space. Define $\sigma(T)=\lbrace \lambda\in \mathbb{C} | \lambda I - T~\textrm{is not invertible}\rbrace$. How can I prove that $\sigma(T^n)=\lbrace \lambda^n|\lambda\in \sigma(T)\rbrace$?
user8484
  • 801
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span density of shift operator

Let $\mathcal{H}$ be a Hilbert space with complete orthonormal basis $\{e_n | n\in \mathbb{N}\}$. Let $R$ be the shift operator on $\mathcal{H}$ defined by $R(e_n)=e_{n+1}$, and extended by linearity and continuity. Show that there exists no $x\in…
am_11235...
  • 2,142
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Isn't $E\cong E^{**}\iff \dim(E)<\infty $ wrong?

Let $E$ a $K$-vector space and denote $E^{**}$ its bidual. Let $$\Phi: E\longrightarrow E^{**}$$ defined by $$\Phi(x)=\left,\quad f\in E^*.$$ A theorem in my course says $$\Phi\text{ is an isomorphism}\iff E\text{ has finite…
user386627
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sequence of continuous linear operators on banach spaces

Let $T_n$ be a sequence of continuous linear operators from a Banach space $X$ to a normed linear space $Y$. Now, for all $x \in X$, $\lim_{n \rightarrow \infty} T_n(x)$ exists in $Y$. Show that the sequence $T_n$ is uniformly bounded. I can…
user58191
  • 477
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How close is Mazur's Separation Theorem and the 2nd geometric form of HBT?

The Mazur's separation theorem, also known as "Geometric Hahn-Banach theorem", is: Mazur's Theorem. Let $K$ be a convex set having a nonempty interior in a real normed linear vector space $X$. Suppose $V$ is a linear variety in $X$ containing no…
João Victor Bateli Romão
  • 784
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A question about the domain of the product of two unbounded operators

As usual, the product $BA$ of two operators is defined on $D(BA)=\{x\in D(A):~Ax\in D(B)\}$. I have some trouble understanding a notion about that. If $A$ and $B$ are two densely defined operator such that Ran$A \cap D(B)=\{0\}$, then does it follow…
Math
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Find the spectrum of a given operator

Let operator $A:L^2[0, 1]\to L^2[0,1]$ be so that: $$Ax(t)= \begin{cases} x(2t), & t\in[0, 1/2]\\\ x(2t-1), & t\in(1/2, 1] \end{cases}$$ The problem is to find the spectrum of $A$. The operator contracts the graphic of $x(t)$ on $[0,1]$ and inserts…
haidaansko
  • 51
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Non-existence of bounded linear functional on weak-type space

I'm studying the functional analysis book of Stein and Shakarchi. I have a trouble in solving some exercises. The question is about showing non-existence of bounded linear functional on weak-type space. Specifically, we define the weak-type space as…
DJCHOI
  • 61
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On the existence of a functional linear $\varphi: L^{\infty}([-1,1])\to\mathbb{R}$

Let $\chi:[-1,+1]\to \mathbb{R}$ be the characteristic function of interval $[0,1]$, i.e. $$ \chi(x)=\left\{\begin{array}{ccc} 1 &\mbox{if} & x\in [0,1]\\ 0 &\mbox{if} & x\notin [0,1] \end{array}\right. $$ I would like to prove that there is a…
Elias Costa
  • 14,658
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Basic functional analysis question: equivalence of norms.

Suppose we have two norms on a vector space such that a linear functional is continuous with respect to one if and only if it is continuous with respect to the other. Show that the two norms are equivalent. Two norms $||.||_1 $ and $||.||_2$ are…
MathCosmo
  • 806
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If ${x_n}$ converges weakly to $x_0$, then for $T \in B(X,Y)$, $T(x_n)$ converges weakly to $T(x_0)$?

Let $X,Y$ be normed spaces. A bounded operator $T: X \to Y$ is arbitrary. $B(X,Y)$ denotes the set of bounded operators. How to prove that if ${x_n}$ converges weakly to $x_0$, then for $T \in B(X,Y)$, $T(x_n)$ converge weakly to $T(x_0)$? I know…
Johnny Chen
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2 answers

The adjoint of a linear functional

Let $l\in \mathcal{H}^*$ be a real linear functional on a hilbert space $\mathcal{H}$. I want to compute the adjoint $l^*$ in $\mathcal{H}$. Using the Riesz-Fischer representation, the unique existence of a $h_0\in \mathcal{H}$ with $l(h)=\langle…
EpsilonDelta
  • 2,079
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1 answer

Various notations in functional analysis.

In most literature on functional analysis the notation for an inner product on an inner product space $H$ is usually denoted $ \langle\cdot,\cdot\rangle:H \times H \mapsto \mathbb{K} $. However, I have noticed that some authors perfers to write it…
Meagain
  • 693
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Seeing $1/x$ as a distribution

I have to show that by defining $$\langle u, f\rangle=\lim_{\varepsilon\rightarrow 0}\int_{-\infty}^{-\varepsilon}+\int_{\varepsilon}^{\infty}\frac{f(x)}{x}dx$$ with $f\in\mathcal{D}(\mathbb{R})$, we have $u\in\mathcal{D}'(\mathbb{R})$ and find…
rom
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