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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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If $\lambda T$ is closed unbounded operator, how about $T$?

Let $T:A\rightarrow A$ be unbounded operator of an abelian,unital C*-algebra $A$. Let $\lambda\in A$, if $\lambda T$ is closed operator, how about $T$ itself? Is it closed? One of the difficulties is there may exist a sequence $a_{n}$ such that it…
Ken.Wong
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Spectral Radius Definition in An Invitation to Operator Theory

On page 243 in 'An Invitation to Operatory Theory' by Y. A. Abramovich, C. D. Aliprantis, the spectral radius $r(T)$ of an arbitrary operator in $\mathcal{L}(X)$ is defined to be the smallest non-negative real number $r$ for which the closed disk…
Billy Bob
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Orthogonal Complements

In a normed space $X$, the distance $\delta$ from a element $x\in X$ to a nonempty subset $M\subset X$ is defined to be $\delta= \inf_{\hat{y}\in M} ||x-\hat{y}||$. My lecture say: is important to know whether there is a $y\in M$ such that $\delta= …
juaninf
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Isomorphism between Hilbert space and normed space

Can a Hilbert space $X$ be isomorphic (not necessarily isometric) to a non Hilbert normed space $Y$? Or, if there is an isomorphism between Hilbert $X$ and normed $Y$, is $Y$ necessarily a complete inner product space? I am really lost here. I can't…
Citizenx05
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Sequence that converges weakly but not strongly

I'm trying to construct a sequence that converges weakly but not in the norm. I assume that in $l_\infty$, $$x_n = (0,\ldots,0,\underset{\substack{n\text{th}\\ \text{position}}}{1},1,\ldots)$$ will do the job. It does not converge to $0$ in the…
bohem
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Explanation of hyperplane proof

I've just looked at this answer from another question regarding the proof that "a hyperplane is closed $\Leftrightarrow$ $f$ is continuous". I understand everything except this part: Taking $y \in E \setminus \ker f$ , if $f$ were not bounded then…
Quotenbanane
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Show set of elements where sequence of linear operators doesn't converge is dense or empty

Let X,Y be normed spaces and $T: X \to Y$ a linear operator. If $(T_n)$ is a sequence of linear operators also from $X \to Y$ then prove $$A := \{x \in X : T_nx \not\to Tx \}$$ is dense or empty. So essentially, I think what I'm supposed to do is…
Michael
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How to show that $y_K< 4?$

$X = C[0,1]$. $$x_n(t) = \begin{cases} nt, & \text{for $0 \leq t \leq \frac{1}{n}$ } \\ 2-nt, & \text{for $\frac{1}{n} \leq t \leq \frac{2}{n}$ } \\ 0, & \text{for $\frac{2}{n} \leq t \leq 1$} \\ \end{cases}$$ Prove that $x_n$ tend to zero weak …
jasmine
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Can there be minimal dense subalgebra for Banach algebra?

Consider a Banach algebra $A$, can there be minimal dense subalgebra? i.e a subalgebra which is dense and it only contain itself and zero as algebra. I am thinking this thing might exists because for finite dimension it is trivial. However for…
Ken.Wong
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weak topology of $X$ and the weak-* topology of bidual of $X$

Given a normed vector space $X$, is the weak topology of $X$ the subspace topology of the weak-star topology of $X^{**}$?
Jay
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Operator norm of integral operator operator, bound from below

If $f:C[-1,1]\rightarrow \mathbb{R} $ is given by $$f(u)=\int_{-1}^{1} (1-t^2) u(t) \ dt .$$ I managed to bound $||f||$ above by $2$ and so want to find a function such that $|f(u)|=2 $ or some sequence such that $|f(u_n)| \rightarrow 2 $. Any…
Anonmath101
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$x_0$ is in the closure of $M$ iff there is no bounded linear functional on $X$

Following is Corollary 3.6 from Barbara Macclaur's Elementary Functional Analysis. Suppose that $X$ is a normed linear space, $x_0 \in X$ and $M$ is a (not necessarily closed) subspace in $X$. Suppose that $d\equiv \hbox{dist}(x_0,M)>0$ where…
Gregoire Rocheteau
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Under What Conditions and Why can Move Operator under Integral?

Given a function space $V$ of some subset of real-valued functions on the real line, linear operator $L: V \rightarrow V$, and $f,g \in V$, define $$ h(t) = \int_{\mathbb{R}}f(u)g(u-t)du $$ Further, assume $h \in V$. Is the below true? …
user80821
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Prove the functional equation $E(\lambda+\mu)=E(\lambda)E(\mu)$

This equation appeared in the Proof of Theorem 10.9 in the book Functional Analysis by Rudin, p252. The Definition of $E(\lambda)$ is: $$E(\lambda)=\sum_{n=0}^{\infty}\frac {\lambda^n}{n!}a^n.$$ Where $\lambda$ is any complex number and $a$ is an…
allen i
  • 359
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$\int_{0}^{\pi}f(x)\cos (nx) dx=0$ for all $n\geq 0$ implies that$f(x)$ is identically $0$

If $f\in C[0,\pi]$ and $f(0)=0$ then $\int_{0}^{\pi}f(x)\cos (nx) dx=0$ for all $n\geq 0$ implies that$f(x)$ is identically $0$ on $[0, \pi]$. I saw this problem has been solved here before .Here is the link . $\int_{0}^{\pi}f(x)\cos nx =0$ for…
simu tiyam
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