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Could someone explain to me using examples and in layman's terms in which ways topology is related to functional analysis?

After taking an UG course in point-set topology it felt like I had a taste of functional analysis; it seemed familiar and easy while I learnt it. If I could get some guidance referring to concrete mathematical examples, that would be helpful.

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  • $\begingroup$ I am not sure if this will help, but one specific similarity we discussed in class yesterday is the definition of continuity. In functional analysis we use the epsilon-delta version, and in topology we use the fact that the preimage of each open set is open. The exact same idea can be thought of in terms of two different "subjects." $\endgroup$ – fullyhip Feb 17 '15 at 22:34
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    $\begingroup$ Functional Analysis is the interplay of Linear Algebra and Analysis. $\endgroup$ – C-Star-W-Star Feb 17 '15 at 22:39
  • $\begingroup$ @Freeze_S Could you throw light on it by using some concrete example? $\endgroup$ – user_1_1_1 Feb 17 '15 at 22:41
  • $\begingroup$ @GoCodes: Yep but please first tell me what you might be looking for. $\endgroup$ – C-Star-W-Star Feb 17 '15 at 22:44
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    $\begingroup$ @GoCodes: Some examples would be the spectral theorem. (Imho, here most things happen between linear algebra, measure theory and complex analysis.) $\endgroup$ – C-Star-W-Star Feb 17 '15 at 23:48
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A great example of an interplay of functional analysis and (algebraic) topology is the Gelfand-Naimark theorem. See https://en.wikipedia.org/wiki/Gelfand%E2%80%93Naimark_theorem.

I wrote a paper on this topic in my algebraic topology class last semester.

The basic idea is as follows:

Take a locally compact Hausdorff space $X$, and consider the set $C_0(X) = \{f:X \rightarrow \mathbb{C}\}$ of continuous functions from $X$ to the complex plane which vanish at infinity. It turns out that $C_0(X)$ is what is called a $C^*$-algebra.

Moreover, for any $C^*$-algebra $A$, we define a character to be a nonzero $C^*$-homomorphism $\phi:A \rightarrow \mathbb{C}$. Define the spectrum of $A$ to be the set $\Phi(A) = \{\phi:A\rightarrow \mathbb{C}\}$ of characters of $A$.

It turns out that $\Phi(A)$ is a locally compact Hausdorff space. Amazingly, can be shown that $X$ and $\Phi(C_0(X))$ are homeomorphic as topological spaces.

Given an element $f \in C_0(X)$, we denote the (necessarily continuous) function $g_f:\Phi(C(X))\rightarrow \mathbb{C}$ to be given by $g_f(\phi) = \phi(f)$

Finally, define the function $G:C_0(X) \rightarrow C_0(\Phi(C_0(X)))$ to be the function sending $f$ to $g_f$. This is called the Gelfand representation. This map is an isometric *-isomorphism, meaning that $C_0(X)$ and $C_0(\Phi(C_0(X)))$ are also isomorphic. See https://en.wikipedia.org/wiki/Gelfand_representation.

One can prove from this that two locally compact Hausdorff spaces $X$ and $Y$ are homeomorphic if and only if $C_0(X)$ and $C_0(Y)$ are isomorphic.

It is quite awesome, and we should note that the Gelfand transformation is in some sense a very generalized Fourier transform. This connects it to functional analysis.

I hope this gives somewhat of a taste where these two ideas connect. I believe that this is very close to category theory as well.

In layman's terms:

Take a special space $X$, look at the set of certain continuous functions of that space $C_0(X)$. $C_0(X)$ has an algebraic structure. Look at the functions which preserve the algebraic structure (homomorphisms) of $C_0(X)$. Call it $\Phi(C_0(X))$. Then we have that $\Phi(C_0(X))$ is homeomorphic $X$.

Moreover, we know that the space $C_0(X)$ is isomorphic to the space of continuous functions of the space of nonzero $C^*$-algebra homomorphisms of the space of continuous functions of $X$. Or, $C_0(X)$ is isomorphic to $C_0(\Phi(C_0(X)))$.

A corollary to this discovery is that given two special spaces $X$ and $Y$, they are homeomorphic if and only if $C_0(X)$ and $C_0(Y)$ are isomorphic (as $C^*$ algebras).

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