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In my lecture notes, the following text appears:

"..We know that if $(X,d)$ is a metric space and $\mathbb K$ is $\mathbb R$ or $\mathbb C$, then $C(X,\mathbb K)$, that is the space of continuous function from $X$ to $\mathbb K$, equipped with the norm of uniform convergence, is a unitary algebra.."

I never came across the term algebra.

What does an "algebra" mean in this context? And what does "unitary algebra" mean?

Don't bother yourself with a thorough explanation or something, I just need an $a$$b$$c$$d$.. list of the conditions that must be satisfied by a space of functions to be called an algebra, and what is the additional condition that makes it a unitary one. I know that the term "algebra" should not merely be a description for certain function spaces, it should describe a general vector space under certain conditions; but currently, I am just interested in what it means in this context and how the conditions are expressed in terms of functions for the purpose of direct application.

Additionally:

When is a subspace called a "subalgebra"?

Again, I can relate to similar terms containing "sub" to guess that a subspace is called a subalgebra if it is an algebra by its own (with the induced operations from the initial space); but I need an $a$$b$$c$$d$.. list of what makes a subspace a subalgebra.

Thank you very much.

Asaf Karagila
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goldenratio
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    why don't you just consult wikipedia instead of asking a long question? roughly speaking it is a ring which is also a vector space, with a compatibility requirement – Mister Benjamin Dover Apr 11 '15 at 22:10
  • @goldenratio: I stopped bothering to think about answering your question as soon as I read "don't bother yourself ..." . Prejudging and proscribing the answers you want to see is counter-productive to getting good answers. You were lucky that you did get some good answers. – Rob Arthan Apr 11 '15 at 22:18
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    @RobArthan I really don't want to bother anyone with any explanation. I clearly asked for a list of conditions that make a function space an algebra. That is all. I said that for a good reason, and apparently the people answering just skipped that. Though I appreciate the effort. – goldenratio Apr 11 '15 at 22:20
  • @goldenratio Does the answer I gave not do that? – wlad Apr 11 '15 at 22:24
  • @goldenratio: perhaps it's an English language issue: "don't bother" can come across as a very negative thing to say in some contexts (it suggests that the speaker wants to control what is important and what is not in a discussion). – Rob Arthan Apr 11 '15 at 22:29
  • @user3491648: your answer doesn't quite make it clear that the function space becomes an algebra when you equip it with the algebraic operations defined pointwise. That extra structure doesn't just fall out of the metric space structure. – Rob Arthan Apr 11 '15 at 22:40

2 Answers2

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An algebra $A$ over a field $\mathbb{F}$ is like $\mathbb{C}$ to the field $\mathbb{R}$.

It's a vector space $V$ over $\mathbb{F}$ and a mapping $\cdot : V \times V \rightarrow V$ linear in both arguments. An example is $n \times n$ matrices, with $\cdot$ being matrix multiplication. The complex numbers I've already said. The quaternions are an example.

An algebra is unitary if it has an $I$ such that $x \cdot I = I \cdot x = x ,\forall x \in A$. The $I$ is called unity. All the examples above are unitary. But if the product operator is defined in such a way that it sends everything to $0$, then it's not unitary.

A space of continuous functions to a field is clearly an algebra because it's a vector space and the product operation is just multiplication of functions, which is clearly linear both left and right.

A subalgebra of an algebra is a vector subspace which is closed under the product operation.

wlad
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If $A$ is a commutative ring, then an (associative, unital) $A$-algebra is a ring $B$ with a ring morphism $A\rightarrow B$ mapping $A$ to the center of $B$. (It is understood that $B$ is endowed with the restriction of scalars $A$-module structure.) The unitary requirement refers to the existence of a multiplicative identity (I take all rings to be unital), so this is implicit in my definition. The morphism $A\rightarrow B$ is called the structural morphism. An $A$-subalgebra of $B$ is a subring which contains the image of the structural morphism $A\rightarrow B$.

Mister Benjamin Dover
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