# 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).

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### Notations on Craig-Wayne's 1993 paper on PDE

I'm currently reading Craig-Wayne's Newton's Method and Periodic Solutions of Nonlinear Wave Equations. My background includes undergraduate functional analysis and PDE, and there are some notations ...
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### Under what hypotheses is $B$ self-adjoint?

Let $A$ and $B$ be two linear operators such that $B$ is closed and $A\subset B$ with $A$ self-adjoint. It is true that $B$ is also self-adjoint? If it is not, what conditions on $B$ guarantee the ...
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### Determine the operator norm of the following function

I have given a function $L: R^2 \to R^2$ , defined as $L(v)= A*v$ where $A$ is a $2\times2$ Matrix \begin{bmatrix} 1 & 0\\ 0 & 2 \end{bmatrix}. Determine the operatornorm and show that a $v$ ...
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### why $x_m$ converges weakly to $x_\infty$?

Let $(X,\|.\|)$ be reflexive Banach space and $Y$ be a closed separable subspace of $X$ $\big((Y ,\|.\|)$is clearly a separable reflexive Banach space$\big)$, then the dual space $Y^*$ of $Y$ is ...
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### Prove an equivalent norm

Consider $g\in C[0,1]$ a strictly positive function and bounded away from zero, prove $\|f\|_g=(\int_0^1g(x)|f(x)|^2dx)^{\frac{1}{2}}$ is equivalent to $\|f\|=(\int_0^1f^2(x)dx)^{\frac{1}{2}}$. I ...
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### Some questions about wave operator.

On sec. 3.4.1 in Schlag & Nakanish: invariant manifolds and dispersive Hamiltionian evolution equations, the authors talked something about wave operators. The wave operators are defined as the ...
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### Find it adjoint $T^{*}$ [closed]

Show that the operator $$T: \ell_{2} \rightarrow \mathbf{C},Tx := \sum_{n=1}^{n=\infty} \frac{1}{n}x_{n}$$ is bounded.Then find it (Banach) adjoint $T^{*}$.
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### What does the notation $A\in\mathscr{B}(H_1, H_2)$ mean?

I am sorry for the trivial question, but I am a little bit confused about this notation in literature. Let $H_1$ and $H_2$ be two Hilbert spaces. I am interested in understanding what means that an ...
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### Which is the spectrum of this operator?

Let $T : \ell_2 \to \ell_2$, $T(x_1,x_2,x_3,...,x_n,...) = (0,\frac{x_1}{1},\frac{x_2}{2},\frac{x_3}{3},..., \frac{x_n}{n},...)$. Which is the spectrum of this operator?
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### Showing that operator is compact

Definition: Let $X, Y$ be Banach spaces. Then we call a linear map $T: X \to Y$ a compact operator if $\overline{T(K_1^X(0))}$ is compact in $Y$. $K_1^X(0)$ denotes the ball with radius $1$ and ...
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### $n^{1/p}\chi_{[0,1/n]}$ DOES NOT converge weakly to $0$

I'm working on this problem "Let $X=[0,1]$ with the usual measure. For $n=1,2,...$, let $f_n=n^{1/p}\chi_{[0,1/n]}$. Prove that $f_n$ does not converge weakly to $0$ in $L^1([0,1])$." Yes, this ...
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### Cardinality of a Hamel basis of $\mathbb{R}^\mathbb{N}$ [closed]

I have the vector space of $\mathbb{R}^{\mathbb{N}}$ with scalars in $\mathbb{R}$. How could I find the cardinal of Hamel's base from $\mathbb{R}^{\mathbb{N}}$?, I can't think of any way to attack ...
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### Coset, Hahn-Banach theorem

I would like to solve the following exercise. $X^*$ denotes the space of bounded linear functionals on the normed vector space $X$. Following the hint, we may consider the coset $\{x\}+Y$ of $Y$ ...
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### Representation of diagonal operator in $l^2$ as a multiplication operator in $L_2(X,\mu)$ (spectral theorem)

There is a (weaker) version of spectral theorem saying that any self-adjoint operator in Hilbert space is unitarily isomorphic to multiplication operator in $L_2(X,\mu)$, where $(X,\mu)$ is some ...
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### Young's inequality to estimate $\int_{\Omega} |u|^{p-2}u v \leq \frac{p-1}{p}\int_{\Omega} |u|^{p} +\frac{1}{p}\int_{\Omega} |v|^{p}$

Let $\Omega \subset \mathbb{R}^{n}$ be a Lebesgue measurable subset and assume $u,v:\Omega \rightarrow \mathbb{R}$ are such that $u,v\in L^{p}(\Omega)$. Let $p>1$. I would like to apply Young's ...
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### On a subspace that is isomorphic to a dense subspace

Let $X$ be an infinite dimensional Banach space and let $M$ be a dense subspace of $X$, i.e., $\overline{M}=X$. Let $N$ be another subspace of $X$ such that $N$ is topologically isomorphic to $M$. ...
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### Resolvent definition: bounded operator vs. unbounded operator

Maybe my question is obvious in some sense, but I ask here because I didn’t find a “satisfactory” answer on the web. If we have a bounded or unbounded operator, the definition of resolvent changes? ...
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### Limit in distributions of $\frac{\sin(tx)}{x}$

How can I find the limit of $\frac{\sin(tx)}{x}$ as $t \to \infty$ in $D'$ ? I understand that i need to see the $\lim_{t \to \infty}{\int_{\infty}^{\infty}{\frac{\sin(tx)\phi(x)}{x}dx}}$ for every ...
### How to construct cyclic vector of diagonal operator $T_\lambda:l_2\to l_2$? [closed]
There is a result saying that diagonal operator $T_\lambda$ is cyclic iff all $\lambda_n$ are distinct. I am looking for an example of cyclic vector of such an operator. Thank you.