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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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Banach Alaoglu Theorem vs Sequential Banach Alaoglu Theorem

The Banach-Alaoglu says that the closed unit ball in a dual space $X^*$ of a normed space $X$ is compact in the weak-star topology. The sequential Banach-Alaoglu says that the closed unit ball in a dual space of a separable normed space is…
Jan Lynn
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Origin of Fredholm operators

It is known that the notion of Fredholm operator, without using this name, was introduced by F.Noether in 1921. Can anone say, who and when proposed to call these operators Fredholm? For a certain time the name 'Noether operator' was also used. Who…
Grigori Rozenblum
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A question about the uniqueness in a version of Riesz representation theorem

In my notes, I have this version of the Riesz representation theorem: Let $X$ be a separable Hilbert space, and let $\{\phi_n\}_n\subset X$ a countable orthonormal set. Let $\{c_k\}_k\in l^2$ a sequence. There exist an element $x\in X$ such that…
user479251
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Necessary and sufficient conditions under which $\|A\|_S=0$.

Let $E$ be a complex Hilbert space, with inner product $\langle\cdot\;, \;\cdot\rangle$ and the norm $\|\cdot\|$ and let $\mathcal{L}(E)$ be the algebra of all bounded linear operators on $E$. For $S\in \mathcal{L}(E)^+$, we consider the following…
Student
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Spectrum and Point-spectrum of an operator

Let $(\lambda_n)_{n\in\mathbb{R}}$ be a bounded sequence. $T:l^p\rightarrow l^p, (Tx)_i:=\lambda_ix_i\forall i\in\mathbb{N}$. Find the spectrum $\sigma(T)$ and point-spectrum $\sigma_p(T)$. The definitions are: …
Tobi92sr
  • 1,661
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Continuous linear operator, $T: (C[0,1],\|\cdot\|_p)\to (C[0,1],\|\cdot\|_p)$ continuous for some $p$

Observe $C[0,1]$ and for $1\leq p<\infty$ the norm $\|f\|_p=\left(\int_0^1 |f(t)|^p\, dt\right)^{1/p}$. Let $T: C[0,1]\to C[0,1]$ be an arbitrary linear operator. Show, that when it exists a $1\leq p<\infty$ such that $T: (C[0,1],\|\cdot\|_p)\to…
Cornman
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Reflexifity of the Banach space of continuous functions on $[a,b]$

I need to show that the space $C[a,b]$ is NOT reflexive. This is a question in which I have already showed a few things. We have $S$ an infinite set and $\{s_n\}_{n=1}^{\infty}$ a sequence of distinct points in $S$. $X$ is a Banach space of…
user428890
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If $X/M$ is a normed space with the induced seminorm, is $M$ closed?

Let $X$ be a normed space. If $M$ is a subspace, then $X/M$ has a known seminorm: $\left\|{x+M}\right\|=\inf\{\left\|{x+y}\right\|:y\in M\}$ It is easy to show that if $M$ is closed then $\left\|{}\right\|$ is a norm in $X/M$ (the only class with…
Tanius
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Determine $\left \| T_n \right \|$ depending on $n$ and $p$

For each $n\geq 1$, let $T_n:\ell_p(\mathbb N)\to \mathbb C$ be the linear functional given by $T_n(x)=\sum_{i=n}^{2n}x_i$, where $x=(x_k)_{k\geq 1}\in \ell_p(\mathbb N)$. Let $p$ be any number in $[1,\infty]$. My problem is to determine $\left \|…
Hopeless
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The range of the self-adjoint operator

I have no idea about this problem. Let $A\in B(H)$($B$ is the space of linear bounded operator,H is the Hilbert space) is self-adjoint and Ker(A)={0},I want to prove that $\overline{R(A)}=H$(R(A) is the range of the operator A)
89085731
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Any bounded sequence in $L^4$ has a convergent subsequence in $L^2$?(True or False)

Is this true that "Any bounded sequence in $L^4[0,1]$ has a convergent sbsequence in $L^2[0,1]$?" Of course I tried so much for this but I could not get any solution till now. Any hints. is appreciated. Thank you
Mini_me
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$\|f\vert_M\|=\|f+M^0\|$, Exercise from Folland's functional analysis chapter

I have a question on functional analysis, on something rather simple, but I'm stuck for good. Here it is. If $X$ is a Banach space and $M
Just dropped in
  • 15,513
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Approximation of support compact functions

Let $\Omega$ be Hausdorff locally compact and let $X$ be a Banach Space with norm $\| \cdot \|$. Denote by $C_c (\Omega, X)$ the space of continuous function $f: \Omega \to X$ with compact support $K_f$. I'm asked to proof that: Given $f \in…
user 242964
  • 1,898
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1 answer

Annihilator of an annihilator

Let X be a normed space and X' the dual space of X. The annihilator of a vector subspace $M\subset X$ is defined by: $$M^{\perp}:=\{f\in X'|f(y)=0 \forall y\in M\}\subset X'$$ Is $N\subset X'$ a vector subset of a dual space, then the annihilator…
Tobi92sr
  • 1,661
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Is the set of increasing function closed in $L^{2}$?

Consider $\mathcal{F}\subset L^2[0,1]$ such that every element in $F$ can be represented by a weakly increasing function (i.e., equals to a weakly increasing function a.e.), then is $\mathcal{F}$ closed? I'm very tempted to say yes but haven't got a…
Ecthelion
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