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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 .

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Is the open mapping theorem about linear maps in Banach spaces dependent on the axiom of choice?

The open mapping theorem is usually proved in most texts using Baires Category theorem which depends upon the axiom of choice. But if one studies differential calculus in Banach spaces say as in Dieuodenne Foundations of Modern Analysis the…
Anil Pedgaonkar
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Different Types of Continuity in Reflexive Banach Space

Let $X$ be a reflexive Banach space with dual $X^*$. Let $K\subset X$ be a nonempty closed convex set. The mapping $F: K\rightarrow X^*$ is said to be: weakly continuous if $F$ is continuous w.r.t. the weak$^*$ topology on $X^*$ and the induced…
blindman
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natural embedding of normed linear space is an isometry

To review for an exam, I'm trying to write up a short proof of the following: Let $J: X \rightarrow X^{**}$ be the natural embedding of the normed linear space $X$ into its bidual $X^{**}$, given by $J(x) = f(x)$. This embedding is a linear and…
user58191
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Is the dual of the direct sum isometrically isomorphic to the direct sum of the duals?

Let $X=V\oplus W$ be a normed space; the title is self-explanatory: is it true that $X^*\cong V^*\oplus W^*$? What I've done is the following: I defined $F:X^*\to V^*\oplus W^*$ as $F(\phi)=(\phi|_V, \phi|_W)$. It is obvious that $F$ is well…
Just dropped in
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What does the integral of a delta distribution even mean?

Formally, we define $\delta(\phi)=\phi(0)$ where $\phi$ comes from a suitable class of test function. Based on this, the expression $\int_{-\infty}^{\infty} \delta(x) dx$ seems completely meaningless and I'm unsure how to attribute meaning to…
user540665
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Why an unbounded operator is not-constructive relying on Hahn Banach theorem?

Let $T: \mathcal S(\mathbb R)\to L^2(\mathbb R)$ defined by $$Tf(x)=f'(x),$$ where $\mathcal S(\mathbb R)$ is the Schwarz space on $\mathbb R$. The question is : is there a continuous extension to $L^2(\mathbb R)$, and the answer is no, because…
user621345
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Geometric Hahn Banach implies Analytic Hahn Banach.

I want to prove that the geometric Hahn Banach theorem implies the analytic one. Edit: To avoid confusion I will state the vesion of H.B theorems im familiar with: Analytic H.B: Let $X$ be a linear space(over $\Bbb R$) and $Y\subset X$ a…
user335501
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Convergence of a series given properties on uniformly bounded functions

I have the following problem from "Bollobas, Linear analysis. An introductory course" Let $\phi_n:[0,1]\to\mathbb{R}^+$ $(n=1,2,\dots)$ be uniformly bounded continuous function such that $$\int_{0}^1\phi_n(x)dx\geq c$$ for some $c>0$. Suppose…
jnaf
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unconditional convergent

I have to decide whether the following statement is true or false : every permutation of a basic sequence is equivalent to the entire sequence ! where a sequence $(x_n)$ in a Banach space $X$ is called basic if it's a basis of…
user61965
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Does the sum of two linear operators with real spectra have real spectrum?

Suppose $C=A+B$, where $A$ and $B$ are linear operators defined on an infinite dimensional Hilbert space with real spectra, but they are not necessarily self adjoint operators. Is it true that $C$ always have real spectrum? Any comment or reference…
ssongnul1
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Why is there no space whose dual is $C_\mathbb{R}[0,1]$?

Possible Duplicate: $C_0(X)$ is not the dual of a complete normed space Is any Banach space a dual space? While studying for a course of functional analysis I read somewhere that there is no normed vector space $X$ with $X^*=C_\mathbb{R}[0,1]$. I…
Daniel Robert-Nicoud
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Show that a subspace is dense in $L^2[0,1]$

Let $f \in L^1[0,1]$, but $f \notin L^2[0,1]$. Consider the subspace $X$ of $L^2[0,1]$ such that $X= \{\phi \in L^2[0,1]: \int f \phi = 0\}$. Want to show that $X$ is dense in $L^2[0.1]$. I tried proving this by checking that $\langle f, g \rangle,…
penny
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Is every absorbing set a neighborhood of zero?

I'm studying functional analysis, currently a chapter about topological vector spaces. It is stated that every neighborhood of zero is an absorbing set. But I was wondering if the reversed statement is also true? Isn't, as an counterexample, in $X…
FeWa
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Intuition behind : for all $x\in E$ there is $f_0\in E^*$ s.t. $\left=\|x_0\|^2$ and $\|f_0\|=\|x_0\|$.

I'm seeing a theorem that say that for all $x\in E$ there is $f_0\in E^*$ s.t. $\left=\|x_0\|^2$ and $\|f_0\|=\|x_0\|$. I recall that $E^*$ denote the topological dual of $E$. Is there a similar result for inner product spaces for…
user380364
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Any two norms equivalent on a finite dimensional norm linear space.

I am trying to understand the proof that every two norms on a finite dimensional NLS are equivalent. I am working with this proof I found on the web: http://www.math.colostate.edu/~yzhou/course/math560_fall2011/norm_equiv.pdf My first question is,…
harajm
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