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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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Proof of inequality (mollifier)

Let $J$ be a mollifier, e.g. a function in $J \in C^\infty(\mathbb R^n)$ with the properties $J\geq 0$ and $\int J(x) \mathrm dx=1$ and $J(x)=0$ for all $x$ with $|x|>1.$ Now define $J_\varepsilon (x):=\varepsilon ^{-n}J(\varepsilon ^{-1}x)$ and…
David75
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Sequence of bounded linear functionals on $C^1[0,1]$ that shows Principle of Uniform Boundedness fails without completion.

Let $X$ be the normed vector space $C^1[0,1]$, of continuously differentiable functions on $[0,1]$ with the sup norm $\displaystyle \|f\|=\max_{t\in[0,1]}|f(t)|$. Find a sequence of bounded linear functionals $T_n:X\to\mathbb{R}$ such that for every…
user3784030
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Positive Linear Functional on $C[0,1]$

I have an exercise which seems to be missing some information. Or it could be that I really don't need that information at all. Please let me know what you think and give a solution if possible. Thank you in advance. "A linear functional $f$ on $X =…
inkievoyd
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Conservation of the weak topology by homeomorphism

I have some questions about Brezis book. We know that M is reflexive, so there exists an homeomorphism $J:(M,\|\|_M) \to (M'',\|\|_{L(M',R)})$ between the "strong" topology of M and M''. (Moreover J is isometric). So i would like to know why…
curious
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projections in normed linear spaces

Let $H$ be a Hilbert space and $C$ be a non empty closed convex subset of $H$ and let $x\notin C$. We know that there exists a unique $y_0$ in $C$ such that $\|x-y_0\|=\inf_{y\in C}\|x-y\|$. Call $y_0$, the projection of $x$ onto $C$. The proof of…
Ashok
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Unique unit normal vector (supporting hyperplane) of strictly convex set

When I try to prove this problem, in $\mathbb{R}^n$, we know that a set $K$ which is closed, convex, then every point $x_0\in \partial K$ admits a supporting hyperplane, i.e there exists a vector $\alpha\in \mathbb{R}^n$ such that $$ \langle y,…
Sean
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Check if the identity matrix is an extreme point of the set $S$

I'm trying to check that $I$ is an extreme point of $S=\{A \in M_{2\times2}:\|A\|_1 \leq 1\}$? I have done this by writing out $I=\lambda B + (1-\lambda)C$ with $B,C \in S$ and $\lambda \in (0,1).$ Then I have written out the system of equations for…
maths1234
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Exercise 1.19 in Brezis' Functional Analysis

I would like a clarification to an exercise in Brezis' Functional Analysis. Exercise 1.19(2) says Let $E$ be a normed vector space. Let $F:\mathbb R \rightarrow (-\infty, +\infty]$ be a convex lower semi continuous function such that $F(0)=0$ and…
John Doe
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Show that $\{x_1,...,x_n\}$, where $x_j=t^j$, is a linearly independent set in the space $C[a,b]$.

Show that $\{x_1,...,x_n\}$, where $x_j=t^j$, is a linearly independent set in the space $C[a,b]$. I think I can use properties of polynomials in $R[x]$ here, but I'm not sure. Using $\sum_{i=1}^n c_i t^i$, $\forall t\in C[a,b]$? Any help would be…
Desperate Fluffy
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Existence of a dense hyperspace

How do you prove that every infinite dimensional normed space contains a dense hyperspace? (Where hyperspace is defined to be maximal proper subspace.)
user172730
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why say "$\mathbb{R}$-tree"?

Definition. An $\mathbb{R}$-tree is a metric space $(X,d)$ such that there is a unique geodesic segment (denoted $[x,y]$) joining each pair of points $x,y \in X$; if $[x,y] \cap [y,z] = \{y\}$, then $[x,y] \cup [y,z] = [x,z]$. why this space…
Sara
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Extreme points of the unit ball of $l^1(\mathbb{Z^+})$ and $L^1[0,1]$

Determine the extreme points of the unit ball of $l^1(\mathbb{Z^+})$ and $L^1[0,1]$. My attempt: I know the definition but I don't know how to find these extreme points.Please help me to solve this problem.Thanks in advance. Extreme point:An…
Flip
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The regularity of Dirichlet form in Besov space

Let $C_{0}(\mathbb{R}^n)$ denote the set of continuous functions in $\mathbb{R}^n$ with compact support, $C_0^\infty(\mathbb{R}^n)$ denote the set of infinite differentiable functions in $\mathbb{R}^n$ with compact support. We…
yangmengqh
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What is the limit of the $n$-th power of the shift operator

Let $T: \ell^2 (\mathbb R) \to \ell^2 (\mathbb R)$ be the left shift operator $(x_1,x_2,x_3, \dots) \mapsto (x_2,x_3,x_4,\dots)$. Let $T^n$ denote a left shift by $n$ positions. What is $\lim_{n \to \infty} T^n$? Is it $0$? Edit: I want to have $T^n…
tom b.
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Difference isometrically isomorphism and homeomophism.

What is the difference between isometrically isomorphism and homeomorphism?is an isometric mapping is continuous?
user18719
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