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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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Positive on a dense subspace implies positive in the whole Hilbert space

Here's my question: Let $H$ be a (separable) Hilbert space. Let $W$ be a dense subspace of $H$ and $P: H \to H$ be a linear operator satisfying $\langle Pw, w\rangle \ge 0$ for each $w\in W$. Then $P$ is a positive operator on $H$. My goal is to…
ashK
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How to prove that withouting using Gelfand spectral radius theorem?

Let $X$ be a Banach space and $T$ be a bounded linear operator on $X$. Assume that there is an integer $n\ge 1$ so that $\|T^n\|<1$. Show that $I-T$ is invertible and $$ (I-T)^{-1}=\sum_{j=0}^\infty T^j $$ How to prove that withouting using Gelfand…
H.Y Duan
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Prove that $\sigma(T) \subset \mathbb{S}$

I look at this question:Spectrum of isometry, but I try to consider the similar question: Let $X$ be a Banach space. Let $T\in \mathbb{B}(X)$. If $T$ is an isometry and invertible, prove that $\sigma(T) \subset \mathbb{S}:=\{\lambda:…
Hermi
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One important question about Schwartz space

I have one problem with Schwartz space. I remind you which definitions which I am using. Multi-index: $\alpha=(\alpha_{1}, \ldots , \alpha_{n}) \in \mathbb{Z}_{+}^{n}$, length of multi-index: $| \alpha | = \sum_{i=1}^{n} \alpha_{i}$, partial…
Marcin
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Let $X, Y$ be Banach spaces. If $T^\prime$ is an isomorphism so is $T$

I spent quite a long time trying to proof this, but I am not 100% sure this proof works, and was hoping if someone could maybe proof read this for me. Let $T^\prime: Y^\prime \rightarrow X^\prime$ the adjoint of $T$, show that if $T^\prime$ is an…
soph6626
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Is the sum of infinite and finite dimensional space isomorphic to the original space?

If we have an infinite dimensional normed space $X$ and a finite dimensional one $Y$, is $X \bigoplus Y$ isomorphic to $X$? I cannot conclude if it is true not get a counterexample. I was thinking that if $dim(Y) = n$ and we take $x_1, \cdots, x_n…
Eparoh
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Prove that $\bar{N} \subset\left(N^{\perp}\right)^{\perp}$

I got the following question: If $N$ is a linear subspace of the dual space $X^{\prime}$, the annihilator of $N$ is defined as $$ N^{\perp}:=\{x \in X: f(x)=0 \text { for all } f \in N\} \subset X \text {. } $$ $$ M^{\perp}:=\left\{f \in X^{\prime}:…
soph6626
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Weak convergence in $\ell^1$ implies strong convergence in $\ell^1$. Understanding proof.

See the suggested proof here. The thing I don't understand is that the functional $f\in(\ell^1)^*$, corresponding to a sequence $y\in\ell^\infty$ depends on the index $j$ in its definition. So let $f_j$. Then David Bowman shows that…
sum_math
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Example of dual pair

In the book Banach space theory of Fabian et. al. appears this example about dual pairs: Now, my question is about why is it necessary to use the Tietze-Urysohn theorem to prove that $C(K)$ separates points of $\text{span}\{\delta_k: k \in K\}$. As…
Eparoh
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Holder Inequality context

I was trying to demonstrate that the space l^p is a metric space. Let $p \geq 1$ a fixed real number. By definition, each element in the space $l^p$ is a sequence $x=(\xi_j)=(\xi_1,\xi_2,...)$ of numbers such…
ends7
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About completeness of $pH+qH$.

I readied a notes of Functional Analysis and I see the follow definition: $$ p\vee q=\overline{pH+qH} $$ where $H$ is a Hilbert space, $p,q\in B(H)$ are projections ($p=p^2=p^*$), more precisely, $pH$ and $qH$ are the image of $p$ and $q$ (resp.).…
Kempa
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Showing that $c$ is complete, by showing it is closed

I am kinda stuck at the idea to solve the problem below. I know that $l^{\infty}$ is a Banach space. Further, I know the following Lemma: Let $X$ be a Banach space and $U$ a closed Subspace, then $U$ is complete. I want to show that the subspace of…
DenisGerhardt
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Finding the Norm of an Element

This may sound very trivial, but I do not know what I am missing. Take $X$ be the space of complex valued continuous functions on $[0,1]$ with the usual sup norm. Take $Y=\{f \in C[0,1]:f(0)=0\}$. Show that: $Y$ is closed in $X$ $X/Y$ is…
Ester
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Decomposition of functions in $(l^\infty(\mathbb(N)))^{*}$, the dual space of bounded sequences.

I have the following problem. Its supposed to be easy but i cant really get to a solution so i would be very thankful if somebody could point me in the right direction. Show that for every $\lambda\in(l^\infty\mathbb(N))^{*}$ (the dualspace of…
NoIdea
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Is this weakened condition equivalent to a failure of local convexity?

I'm not quite sure if this is true, and I'm thinking about whether I can weaken the assumptions needed for local convexity in a TVS $V$. My question is this: Suppose there exists a neighborhood $U$ of $0$ such that for all $W \subseteq U$ such that…
Subhasish Mukherjee
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