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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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Let $e,f$ be unit vectors in a real Banach space s.t $\|2e+f\|=\|e-2f\|=3$, show that $\|\lambda e+\mu f\|=|\lambda|+|\mu|$.

Let $e,f$ be unit vectors in a real Banach space s.t $\|2e+f\|=\|e-2f\|=3$, show that $\|\lambda e+\mu f\|=|\lambda|+|\mu|$. I have a hint which is to show there is are linear functionals of norm $1$ s.t $\phi(e)=\phi(f)=1$ and $\pi(e)=\pi(-f)=1$. I…
2132123
  • 1,565
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Closure of range of injective compact operator on a Hilbert space

Let $\mathcal{H}$ be a separable Hilbert space and $C$ a compact operator on $\mathcal{H}$. Assume that $C$ is injective. Is it then true that the closure of the range of $C$ is $\mathcal{H}$?
Lundborg
  • 1,646
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Let $X$ be a normed vector space. Let $T,S$ be bounded linear operators such that $T^2=T,S^2=S,ST=TS$. Show either $T=S$ or $\|T-S\|\geq 1$

Let $X$ be a normed vector space. Let $T,S$ be bounded linear operators such that $T^2=T,S^2=S,ST=TS$. Show either $T=S$ or $\|T-S\|\geq 1$. My observations: $1\leq\|S\|,1\leq \|T\|$ $\|T-S\|=\|T^2-S^2\|\leq\|T-S\|\|T+S\|$ Thus $1\leq \|T+S\|$ I am…
2132123
  • 1,565
4
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Proof check about bounded operator

Let $X$ and $Y$ be Banach spaces, and fix a bounded linear operator $A \in \mathcal{B}(X, Y)$. Choose $\mu \in Y^{*}$, and define a functional $A^{*} \mu: X \rightarrow \mathbf{F}$ by $\left(A^{*} \mu\right)(x)=\mu(A x)$, for $x \in X$. I want…
Maskoff
  • 607
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Why is the interior of $\{(x_n)_n \in {\displaystyle \ell ^{2}}| \sum_{n=1}^{\infty}n^2|x_n|^2 < \infty\}$ empty?

Let $M=\{(x_n)_n \in {\displaystyle \ell ^{2}}| \sum_{n=1}^{\infty}n^2|x_n|^2 < \infty\}$. How can I show that the interior of M is empty? I already showed that M is convex but I'm not sure if that's helping.
Lea
  • 53
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bounded self-adjoint operator has non-empty spectrum

Show that the spectrum of a bounded self-adjoint linear operator on a complex Hilbert space $H\neq\{0\}$ is not empty. If possible, let the spectrum $\sigma(T)=\emptyset$. So its resolvent set $\rho(T)$ equals $\mathbb{C}$. So for all…
am_11235...
  • 2,142
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Simple proof that $z\mapsto \exp(i\langle \omega,z\rangle)$ are linearly independent

I am interested in proving that the family of functions $$\{f_{\omega}: \mathbb{C}^n\rightarrow\mathbb{C}, f_\omega(z) = \exp(i\langle \omega, z \rangle): \omega \in \mathbb{C}^n\},$$ where $\langle \cdot,\cdot\rangle$ is the usual hermitian dot…
user70018
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A problem in open mapping theoram from Kreyszig Section 4.12 Problem 6

I am studying Functional analysis from Kreyszig book and can somebody please help with this problem Problem: Let $X$ and $Y$ be Banach Spaces and $T : X \to Y$ be an injective bounded linear operator. Show that $ T^{-1} : \mathscr R(T) \to X$ is…
user775699
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1 answer

A proof with Legendre polynomials and an integral minimum value

I need to prove that, over the monic polynomials $f$ of degree $n$, the integral $$\int_{-1}^1\bigl(f(x)\bigr)^2\,dx$$ takes its minimum value when $$f(x)=\frac{2^n}{\binom{2n}n}L_n(x),$$ where $L_n(x)$ is the $n$th Legendre polynomial.
deiota
  • 67
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Functional Analysis. Continuous extensions of linear functionals.

I would like to show that a continuous functional on a subspace $W$ of a normed space $(V,\|\cdot \|)$ has a unique continuous extension to $V$ iff $W$ is dense in $V$. I have proved $(\Leftarrow$). But the converse is currently eluding me! Help…
user58514
4
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3 answers

How can I show that it's a Banach space?

Let $I=[a,b]$ (where $a
Charles Seife
  • 563
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SOT limit of Self-adjoint operators is self-adjoint?

Say $T_n$ is a sequence of self-adjoint operators on a Hilbert space and converges in the strong operator topology to $\mathcal{T}$, must $\mathcal{T}$ be self-adjoint? Since $T_nx$ converges to $\mathcal{T}x$ in norm, it converges weakly, and so I…
user39992
  • 540
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A point such that can't be "stably" approximated

Let $T:X\to Y$ be linear and continuous between Banach spaces $X,Y$. Suppose that $\overline{T(X)}=Y$ and $T(X)\neq Y$. Claim: there exists a $y\in Y$, such that for every $x_n$ with $Tx_n\to y$, $$\|x_n\|\to \infty,\quad n\to…
user515599
4
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1 answer

Dual Space of space of weighted functions

Let $g: \mathbb{R} \to \mathbb{R}$ be a positive function uniformly bounded away from $0$. Let $C(\mathbb{R})$ be the space of continuous functions that with norm $| f | := \sup_{x \in \mathbb{R} } |f(x)|/ g(x)$, so it contains all bounded and…
Victor
  • 113
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2 answers

Norm of the operator $Tf=\int_{-1}^0f(t)\ dt-\int_{0}^1f(t)\ dt$

Consider the operator $T:(C[-1, 1], \|\cdot\|_\infty)\rightarrow \mathbb R$ given by, $$Tf=\int_{-1}^0f(t)\ dt-\int_{0}^1f(t)\ dt,$$ is $\|T\|=2$. How to show $\|T\|=2$? On the one hand it is easy, $$\begin{align} |Tf|&=\left|\int_{-1}^0f(t)\…
PtF
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