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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 .

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Interpretation for the Functional Determinant

Let $S:V \rightarrow V$ be a linear operator on the function space $V$. It is possible to define a functional determinant for $S$ via the zeta function regularization process. In specific we define first the associated $\zeta$-function…
Giovanni De Gaetano
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Topics of advanced functional analysis

I did a course in introductory functional analysis and liked it. Now I want to learn more functional analysis with the goal of maybe eventually doing research (phd). I tried to find out what (advanced) functional analysis is about here but the list…
Student
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Example of a linear onto map which is not open

We know that every linear open map between normed spaces is onto. This fact actually motivates the Open Mapping theorem which gives extra assumptions for converse to hold true. But I am unable to construct counterexample for the first fact, i.e. I…
ogirkar
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Continuity of $\int_{0}^{1} \frac{f(t)^2}{\sqrt{t}}dt$ wrt Lp norm

Let $F:C^{0}[0,1]\rightarrow \mathbb{R}$, $F(f)=\displaystyle\int_{0}^{1} \frac{f(t)^2}{\sqrt{t}}dt$ How to prove that $F$ is continuous wrt the metric induced by $L^{p}$ norm for $2
George_K
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The weak $\sigma$-$(V,V')$ topology on a normed space V is Hausdorff?

The weak $\sigma$-$(V,V')$ topology on a normed space V is Hausdorff? So it is claimed that The weak $\sigma$-$(V,V')$ topology on a BANACH space V is Hausdorff? In my proof I used the Hahn-Banach theorem to find the existence of linear function…
user58514
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Closed doesn't imply complete

Let $X$ be a norm linear space. If $X$ is banach space then subspace $Y$ is closed iff $Y$ is complete. But if $X$ is not banach space then $Y$ is closed need not imply $Y$ is complete. Can u give me such an example? I took a non-banach space…
Pradip
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Does Closed Graph imply Closed Range

Suppose I have a bounded linear operator from a space $X$ to $Y$, both Banach. I know that if D(T) and Ran(T) are closed, then the graph G(T) is closed in $X\times Y$. However is the converse true ? Does G(T) closed imply that Ran(T) is closed…
me10240
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Prove there exists an unbounded linear operator $T:\ell^1\rightarrow \mathbb R^2$..

How to prove there exists an unbounded linear operator $T:\ell^1\rightarrow \mathbb R^2$?
PtF
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Banach-Steinhaus variant

Let $T_n$ be a sequence of continuous linear operators from a Banach space $X$ to a normed linear space $Y$. Now, for all $x \in X$, $\lim_{n \rightarrow \infty} T_n(x)$ exists in $Y$. Define $T(x) = \lim_{n \rightarrow \infty} T_n (x)$ on $X$.…
user58191
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Characterization of extreme points of the ball of $B(H)$

The following is an exercise in Chapter 5, Section 7 of Conway's Functional Analysis text: If $\mathcal{H}$ is a Hilbert space, show that $T$ is an extreme point of $\text{ball }\mathcal{B}(\mathcal{H})$ if and only if either $T$ or $T^*$ is an…
user631658
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Proving James' Theorem

I am reviewing the proof of James theorem, i.e. a Banach space is reflexive iff every continuous linear functional obtains its norm. Every thing I find online shows one direction ($\Leftarrow$), but not ($\Rightarrow$). I am having issues seeing it…
Scott
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Bounded operator to a space with two different norms

$X$ is a Banach space and $Y$ is a normed linear space. $(Y,\lVert\cdot\rVert_1)$ is not complete and $(Y,\lVert\cdot\rVert_2)$ is complete, while $\lVert\cdot\rVert_2\ge\lVert\cdot\rVert_1$. Let $T: X\to( Y,\lVert\cdot\rVert_1)$ be a bounded linear…
Zeng
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Functional derivative of integral with boundary contribution

What is the functional derivative of a functional $F$ that is expressed as a volume integral over a region $\Omega\subset\mathbb R^3$ plus a surface integral over the boundary $\partial\Omega$? An example for such a functional is $$ F[c] =…
David Zwicker
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Characterization of bounded set in test function space?(Rudin)

Let $E$ be a subset of test function space $D(\Omega)$, then Rudin (functional analysis) in Page 178 says that $E$ is bounded if and only if for any $f\in D'(\Omega)$ , the supremum sup$\{|f(\phi)|:\phi\in E\}$ is finite. How to prove that the…
stephenkk
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Can C([0,1]) be written as a countable union of compact sets?

Let $X=C([0,1])$ be the Banach space of continuous real valued functions on $[0,1]$ (with the $\sup$-norm). I am wondering if $X$ can be written as a countable union of compact sets $K_1 \subset K_2\subset K_3 \dots$?
Cantor
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