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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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Definition of norm

This is probably a simple one. Defining a normed vector space over a field K we ask the norm function to satisfy the equality: $||\alpha x||=| \alpha | \ ||x||$. However, if K is not a field of reals or complex numbers, it is unclear what…
user1928091
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Just a sanity check in basic functional analysis

Consider the algebra $C(S^1)$ of continuous functions $S^1 \to \mathbb C$ together with the $\|\cdot\|_\infty$ ($\sup$-norm). I am thinking that: (?) The (sub-)algebra generated by $\rm{id}$ and $\overline{\cdot}$ (the complex conjugation) is the…
student
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A convex symmetric set in a real vector space is balanced

Show that a convex set in a real vector space is symmetric if and only if the set is balanced. For the backward direction, i.e. if the convex set is balanced in real vector space, then that it is symmetric is easy to show. For the forward direction,…
Idonknow
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Why is this unbounded linear functional also has a closed kernel?

On the space $l_0$ of sequences with finite non-zeros, define $f:l_0\to l_0$ to be: $$f(x)_k=kx_k,$$ then $f$ is famously unbounded. According to my textbook, that means $f$ must have a non-closed kernel. But in this particular case, isn't it that…
xzhu
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Closed ideal in algebra of continuously differentiable complex functions on $[0,1]$

Let $A$ be the banach algebra of continuously differentiable complex functions on $[0,1]$ with pointwise multiplication normed by $ ||f||=||f||_{\infty}+||f'||_{\infty}. $ I have to show that the ideal $ J= \{f\in A : f(p)=f'(p)=0\} $ is closed. Is…
simon
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Properties of $||x||_X\leq c||Tx||_Y+||Kx||_Z$ for every $x\in X$

Let $T: X\to Y$ be a bounded linear operator between Banach spaces.Assume that there exists c$\in R$ and a compact operator $K: X\to Z$ such that $$||x||_X\leq c||Tx||_Y+||Kx||_Z\quad \forall x\in X$$ Prove that a)Ker(T) ha finite dimension b) if…
user62138
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A proof about a proposition on Fréchet spaces

Here is the proposition: assume that $V$ and $W$ are both Fréchet spaces, $u$ is a continuous map from $V$ to $W$. If we suppose that $u$ is injective, and $dim_C$$W/u(V)$ is finite, then $u(V)$ is a closed subspace of $W$. I thought out a proof,…
Cao
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Product and sum of positive operators is positive

I want to show that for $S,T\in B(H)$ bounded operators on Hilbertspace with $S\geq 0,T\geq 0$ and $ST=TS$, we have $S+T\geq 0$, and $ST\geq 0$. $T\geq 0$ means $(Tx,x)\geq 0$. To me it seems that $((S+T)x,x) = (Sx,x)+(Tx,x)\geq 0$. But for $ST$…
DinkyDoe
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Prove an inequality in H1

I need prove that for $n\geq 3$ there exists a constant C such that $\displaystyle\int_{\mathbb{R}^n}\dfrac{u^2}{|x|^2}dx\leq C\int_{\mathbb{R}^n}|\nabla u|^2dx$ for any $u\in H^1(\mathbb{R}^n)$. The hint is: note that $0\leq \left|\nabla…
yemino
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Defining means on closed, unital subalgebras of $\mathscr{C}_{b}(X)$

In what follows, $X$ is a topological space and $\mathscr{C}^{b}(X)$ is the space of bounded, continuous functions on $X$ (with values in $\mathbb{C}$). I give you an adaptation of the definition of a mean as stated in Reiter, H. - Classical…
MWL
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Composition of convergent sequences of continuous functions

Let $f_n \rightarrow f$ and $g_n \rightarrow g$ in $C([0,1])$. Is it true that $f_n \circ g_n \rightarrow f \circ g$? where the metric space we're interested in is $(C([0,1]), d)$ and $d$ is the sup norm: $$ d(f,g) = \sup_{x \in [0,1]} \big|f(x) -…
fxxer
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Character theory questions

I guess that the characters on B(H) are as the form $\phi_x$ where $\phi_x(T)=\langle Tx,x\rangle$. I couldn't prove that the function is homomorphism!
F9119
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$C[0,1]$ endowed with integral norms

Consider the following norms on $C[0,1]$: for each $p\in [1,+\infty]$ $||x||_p=(\int_0^1|x(t)|^pdt)^{1/p}$ (this $p$-norm is induced from $L^p$ wich contains $C[0,1]$). Are they equivalent? Of course, $||x||_p$ is not equivalent to $||x||_{\infty}$…
user74574
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2 answers

$\langle Tx, x \rangle = 0$ gives $T = 0$ in a complex vector space

$T$ is a continuous linear operator on complex vector space with $\langle Tx,x \rangle = 0$ for all $x$. Then $T = 0$. How to prove it? One related question is here, without applying continuity. Some hints will be sufficient. Thank you.
Supriyo
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Norm of a Kernel Operator

So, I was practicing some problems and considered the space $X = C[a,b]$ with the $L_{1}$-norm. I consider the operator $$Tf(x) = \int_a^b k(x,y)f(y)\,dy$$, where $k(x,y)$ is continuous in both of its variables. So, I find this operator is…
user102908
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