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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 $B_X$ is weakly compact, then it is $w^{*}$-closed in $X^{**}$

Suppose that $X$ is a Banach space, $X^{*}$ is it's dual space, $X^{**}$ is its double dual space, and $w$-topology means weak topology while $w^{*}$-topology means weak* topology. Why is the given statement true? I found it on page 75 of Fabian's…
Hugh Mungus
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Inequality in the space of bounded linear operators in Hilbert spaces

Let $(a_n)_n$ be sequence of bounded linear operator on a Hilbert space $E$ and $b$ be a positive operator on $E$, Why $$\left\|\displaystyle\sum_{n=1}^da_n^*ba_n\right\|\leq\|b\|\left\|\displaystyle\sum_{n=1}^da_n^*a_n\right\|\;??$$ Thank you for…
Student
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Describe dual space of $C[0,1]$

I am stuck in this problem. Describe the dual space of $C[0,1]$, where $C[0,1]$ is the Banach space of all real continuous functions on $[0,1]$ induced norm $$ \|x\|_{\max}=\sup_{t\in [0,1]}|x(t)|\quad \forall x\in C[0,1]. $$ Thank you for all…
blindman
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Question on Hahn-Banach theorem

I'm having difficulties with solving following problem. Find a linear functional in $C[0,1]^*$ such that cannot be extended to $L^p [0,1]$ Why such a linear functional cannot be extended? I know I should choose a linear functional which is not…
Detectives
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Example of a strictly positive bound function with constant norm that in mean converges to zero

It is well known that $\int_{0}^{1}\sin(nx)dx\rightarrow 0$ but $\int_{0}^{1}\sin^{2}(nx)dx$ converges to a positive constant. Is there a sequence of strictly positive, uniformly bounded functions with the same property? That is $\int_{0}^{1}…
Akshay Shanker
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Showing that open convex subspace of $L^p[0,1]$ is either empty or $L^p[0,1]$ itself.

I have a question regarding the proof of a proposition. Basically I tried a method, and it seems to work, I don't find a mistake. However I don't use all the conditions in the hypothesis, leading me to believe that there might be a mistake. The…
K.A.
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Is $l^2$ dense in $(l^{\infty}, \|\cdot\|_{\infty})$?

Here I am using $l^2$ to define the set of complex-sequences which are square summable and $l^{\infty}$ to define the set of bounded complex sequences. I think not. If we take $x(n) = 1 \;\; n\in \mathbb{N}$ ,which is bounded sequence. Then for…
user197848
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extreme points of unit balls on $L_1(\mu)$

I wanna show that for a measure space $(X, \mu)$, the set of extreme points of the unit ball of $L_1(\mu)$ equals the set of characteristic functions of atoms of $\mu$ multiplied by a suitable constant(to guarantee that the norm is 1). I tried to…
CSH
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Spectrum of $ T(x_1, x_2, x_3, x_4, \ldots )=(-x_2, x_1, -x_4, x_3, \ldots). $

Find the spectrum of the operator $T: l^p\to l^p$, $1\leq p\leq \infty$ $$ T(x_1, x_2, x_3, x_4, \ldots )=(-x_2, x_1, -x_4, x_3, \ldots). $$ My attempt I have that $Tx = \lambda x$ $$ \implies \lambda (x_1, x_2, x_3, x_4, \ldots )=(-x_2, x_1, -x_4,…
na1201
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Norm on $D(A)$ where $A$ is a unbounded linear operator

Let $E,F$ Banach spaces. Brezis calls a operator $A: D(A)\subset E\to F$, where $D(A)$ is a subspace such that $\overline{D(A)} = E$ of unbounded linear operator. In some exercise he gives the following hint: Consider $D(A)$ with the graph norm. In…
L.F. Cavenaghi
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At least one of $\|S\|$ or $-\|S\|$ of a Self-Adjoint operator must lie in the spectrum. Don't understand the proof.

I'm having a bit of difficulty following the proof. It would be great if someone could give me a bit of advice. I get the feeling that my question is rather trivial so sorry in advance. Many thanks. For $\mathcal{H}$, a complex Hilbert Space and…
mark
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There is no inclusion between $L^p[0,+\infty)$ and $L^q[0,+\infty)$

I want to show that there is no inclusion between $L^p[0,+\infty)$ and $L^q[0,+\infty)$, for any $1\le p
soap
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Redundant condition? What if the image is proper and dense?

Let $X$ and $Y$ be Banach spaces. A bounded operator $T:X\to Y$ is a Fredholm if The dimension of $\ker(T)$ is finite, The codimension of the image $\mathrm{im}(T)$ is finite, The image $\mathrm{im}(T)$ is closed in $Y$. I've found notes saying…
Otavio Kaminski
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Weak star closedness in dual space

Let $(X, \|.\|)$ be a Banach space and $(X^*, \|.\|_{X^*})$ its dual space. Suppose that $E^*$ is convex and closed in the norm topology of $X^*$. Suppose that $X$ is not reflexive, I would like to ask whether $E^*$ is weak$^*$ closed in $X^*$.
blindman
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Can every metric on a vector space be obtained from a norm?

I have a problem understanding the example 2.2-8 from Kreyszig's "Introductory Functional Analysis with Applications". The example shows a metric $d(x,y)$ on the space $s$ - set of all (bounded or unbounded) sequences of complex numbers: $$ d(x,y) =…
Konstantin
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