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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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Hahn Banach extension of the following functional.

The norm on $\mathbb{R}^2$ is defined as follows: $$ \| (x,y) \|=|x|+|y|. $$ Now let $$X=:\{ (x,x):\ x\in \mathbb{R}\}$$ be a subspace of $\mathbb{R}^2. $ Let $ f:X\to \mathbb{R}$ defined as $f(x,y)=3x,\ \forall (x,y)\in X.$ Let $g$ be an extension…
Sachchidanand Prasad
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Why $C(0,1)$ is not complete?

I understand that $C([0,1])$ of continuous complex-valued functions on $[0,1]$ is complete wrt sup norm. But why the space $C((0,1))$ of continuous complex-valued functions on $(0,1)$ is not complete?
Kenneth.K
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problem on completeness of a subspace of C[0,1]

I am really stuck in the following problem: Let $X=C[0,1]$ with the inner product $\langle x,y\rangle=\int_0^1 x(t)\overline y(t)\,dt$ $\forall$ $x(t),y(t)\in C[0,1]$ $X_0 =\{x(t) \in X :\int_0^1 t^2x(t)\,dt=0\}$and $X_0^\bot$ be the orthogonal…
Abhishek Shrivastava
  • 854
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Constructing non-Cauchy sequences in an infinite-dimensional space

Some proof I stumbled on uses that In an infinite-dimensional space we can find a sequence $(x_n)_{n\geq 1}$ such that $\Vert x_n\Vert \leq 1$ and $\Vert x_n-x_m\Vert\geq 1/2$ for all $n\neq m$. I am new to infinite-dimensional spaces. Could…
user161518
  • 197
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The space $L_{2}(D_{1})$ of all analytic functions on $D_{1}$

let $D_{\rho}$ denote the closed disk of $\mathbb{C}$, $|Z-Z_{0}|\leq \rho$ ,$\rho >0$ , $z_{0} \in \mathbb{C}$ are fixed . Assume that $f$ is analytic on $D_{\rho}$ with taylor series : $$f(z)=\sum_{n=0}^{\infty} a_{n} (z-z_{0})^n$$ Prove that the…
Mahmoud Hassan
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A finite dimensional normed space

I would like to find a short proof for the following theorems: Theorem 1. A normed space is finite dimensional iff all of its linear functional is continuous. Theorem 2. A normed space is finite dimensional iff its unit ball is compact. Thank you in…
blindman
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Provide an example of a Banach space $X$ and a functional $f$ in $X'$ such that there is no unit norm vector $x$ in $X$ satisfying $|f|=f(x)$

Give an example of a Banach space $X$ and a functional $f$ in $X'$ such that there is no unit norm vector $x$ in $X$ satisfying $|f|=f(x)$.
user401122
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Show that $\int_{a}^{b}x(t) t^2 dt$ is bounded linear functional and compute it's norm

In our class of functional analysis, we only studied theory, but never did exercises. We haven't studied Lebesgue integration. Also I'm having problems with key understanding of the subject, please don't judge me for that. Let $A : C[a,b] \mapsto…
shcolf
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Continuous map from $l^{\infty}$ to $l^{2}.$

Let $l^{\infty}=\{(a_{n}):a_{n}\in\mathbb{C},sup_{n}|a_{n}|=\|a_{n}\|_{\infty}<\infty\}$ and $l^{2}=\{(a_{n}):a_{n}\in\mathbb{C},(\sum|a_{n}|^{2})^{1/2}=\|a_{n}\|_{2}\}$. Define a map $T:l^{\infty}\rightarrow l^{2}$ as…
neelkanth
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Denseness in $C_b([0,1])$

Let $X_n$ be the set if functions $f$ such that there exists $0\leq s_0
tubmaster
  • 708
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property of separable Banach space?

how to prove: If $X$ is a separable Banach space, then there exists a sequence $\{f_n\}_{n\ge 0}\subset X^*$ such that $$\|x\|=\sup_n |f_n(x)|$$ for all $x\in X$ I don't know how to deal with "separable" since it is a property of $X$, not $X^*$,…
Lookout
  • 2,161
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Sequence of Preimages goes to infinity for bounded linear operator

Let $X$ and $Y$ be Banach spaces and let $T:X\to Y$ be a bounded linear operator such that $T(X)$ is dense in $Y$ but not equal to $Y$. Show that there exists some $y\in Y$ such that each sequence $(x_k)\subset X$ with $Tx_k\to y$ has the property…
sranthrop
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If $Y \subset X$ is dense subspace, then why we can think $Y^*$ and $X^*$ is equivalent?

I saw "If $Y \subset X$ is dense subspace, then we can think $Y^*$ and $X^*$ is equivalent. In fact for $f \in X^*$, $f|_Y \in Y^*$ and the map $ f\to f|_Y$ makes them equivalent" in a book. But I cannot prove this. I think it means that for $T:…
M. Doe
  • 233
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Riemann integral of operator valued function on Hilbert Space

Let $\mathcal{H}$ be a Hilbert space and $F:[a,b]\to B({\mathcal{H}})$ be a continuous function. Suppose $P$ is a spectral measure on $\mathcal{B}([a,b])$. Suppose $F(t)P(\Delta)=P(\Delta)F(t)$ for all $t\in [a,b]$ and $\Delta\in\mathcal{B}([a,b])$.…
Ribhu
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Doubt in the iteration argument on Fredholm theorem's proof.

In final of page 161 here: By assumpuption I know that $E_1 \neq E$, because you're supposing that $I-T$ isn't surjective. However, I didin't understand why $E_2 \neq E_1$ (since $I-T$ is injective.)?
user29999
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