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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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Spectral decomposition, Sturm-Liouville

Consider $A:L^2([0,1]) \to L^2([0,1])$ defined as $$ (Af)(s) = \int_0^1 \max\{s,t\}\cdot f(t)\mathrm dt$$ I've shown that the operator is compact and self-adjoint. Looking for the eigenvalues. $$(Af)(s) = cf(s) = \int_0^1 \max\{s,t\}\cdot…
superAnnoyingUser
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Density of a kernel of a linear functional

Let $X$ be subset of of $L^2(\mathbb R)\cap L^1(\mathbb R)$, dense in $L^2(\mathbb R)$. Let $Tf(x)=\int_{\mathbb R}f(x)dx$ with $f\in X$. Can we say that $\ker T=\{f\in X:Tf(x)=0\}$ is dense in $L^2(\mathbb R)$? One way of proving or disproving this…
Math
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Projective tensor product of the normad spaces

How can we show that if $X$ and $Y$ are infinite-dimensional normed spaces, then the projective tensor product of $X$ and $Y$, $X\otimes_\pi Y$, is an incomplete space? Ps: It is Exercise $2.5$ from Ryan's book (Introduction to tensor products).…
user575596
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Why do we denote the sequence space as $l^\infty$

My question is really simple. I didn't understand the call the sequence space as $l^\infty$. I know this is from the $l^p$ spaces denotation but I didn't understand the connection between both. $l^p$ space Let $p\ge 1$ a fixed real number. By…
user42912
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Exposed points of a compact convex set.

Suppose that $K$ is a compact convex set in a topological vectorial space locally convex such that $K$ is the closed convex hull of its exposed points $(x_\mu)$. I would like to know if it is true that there exists a set of functionals $(f_\mu)$,…
user518824
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Hahn-Banach extension uniqueness.

Given $f:C[0, 1]\to\mathbb{R}$ - a linear function, defined on 1d linear span with $x(t)=t+1$ as basis. Find out if there is unique Hahn-Banach extension preserving norm on the whole $C[0, 1]$. I have tried to recall to Riesz representation…
haidaansko
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bounded linear functionals on normed vector spaces

Let $x$ be in the normed vector space $X$. Then there is a bounded linear functional $f$ on $X$ such that $$f(x) = ||f|| ||x||.$$ Are there other proofs of this proposition, aside from the proof using Hahn-Banach to extend a function on the subspace…
user58191
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Exponential of an Element in a Banach Algebra, and Applying Liouville's Theorem

I am currently reading through Chapter 15 of "Complete Normed Algebras" by F.F. Bonsall and J. Duncan. I have come across the following proposition on page 76: Proposition 5 Let $ A $ be a complex Banach algebra with unit, $ M $ a unit linked…
LMW
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When to use $\|x-y\|$ or $|x-y|$ or $d(x,y)$?

I am having a hard time understanding when to use $\|x-y\|$ or $|x-y|$ or $d(x,y)$? I thought that $\|x\|$ was used for matrices or vectors, $|x|$ was used for scalars, and $d(x,y)$ was used for a metric space? Are there other restrictions for what…
MathIsHard
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inclusion of duals in functional analysis

Let $E$ be a normed space and $F$ a subspace of $E$. Is there any linear mapping $T: F' \to E'$ such that $||T(\phi)|| = ||\phi||$ for every $\phi \in F$? Here is what I am thiking. Take $p < q$, Hölder conjugates. Then, there is the natural…
Brisão
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Criterion for the Existence of a Bounded Left Approximate Identity

I am reading through the text "Complete Normed Algebras" by F.F. Bonsall and John Duncan and I'm having some troubles understanding proposition 2 in chapter 1, section 11 (page 58) regarding the existence of a bounded left approximate identity for a…
LMW
  • 445
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what math branch deals with functional space where composition of the functions are operations?

what is the math branch that research functional space where composition of the functions are operations? Hilbert space and other linear functional space mainly research on the addition and scalar multiplication of two functions in the space. such…
martingale
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Prove that $P_W+P_V$ is orthogonal projection iff $V\perp W$ in Hillbert space.

Let $P_W,P_V$ be orthogonal projections in Hilbert space $\big(H,\langle\cdot, \cdot\rangle\big)$. Prove that $P_W+P_V$ is orthogonal projection iff $V\perp W$. There is hint to consider $(P_W+P_V)^2$ and notice that if $P_V(x)=-x$, then $x=0$ I am…
SekstusEmpiryk
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$\overline{\{Ax:\|x\|\leq 1\}}\subset \{Ax:\|x\|\leq 2\}$

Continuing with the previous question, I want to ask one more question which seems true but I don't have any idea again. If $A\colon X\to \mathbb{C}^{n}$ is a finite rank bounded linear operator, then does the following…
Seewoo Lee
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The proof of Open mapping principle (Lax)

I am reading the proof in Peter Lax book (page 169). By baire category we have some set $MB_i$ dense in some open set U. Then they translate those to origo by the form $M(B_n -x_0)$. And state that $B_n - x_0$ is inside the ball of radius $n +…
Johan
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