How to define a vector with automatically added symbolic elements with subscripts

Question

I want to define a vector lets say k containing elements Symbolize[Subscript[k, 1]], k2, k3 ... etc. When I define lets say k4 . I want the symbol automatically added to vector k. Something like below but not working at all.

k /: {k, Symbolize[
 ParsedBoxWrapper[
 SubscriptBox[
 StyleBox["v", FontFamily -> "Courier New", FontWeight -> "Plain", 
   FontSlant -> "Italic"], "_"]]]} := 
AppendTo[k, Subscript[v, _]];

My Question is;

1) How can I symbolically define elements (with sub scripts) automatically added to a vector.

2) If I assign values to symbols, How can I get elements values only and symbols only.

EDIT 1

Symbolize[Subscript[k, _]] 

Command Symbolizes all subscripts of k e.g. k1 (Subscript[k, 1]), k2 (Subscript[k, 2]) etc. This is fine. What I want is collect k1, k2 k3 in a set (List) k with out additional commands or additional functions. Just as soon as sybolize sub scripted ones it should add it self to set k (sort of pattern maching as in Subscript[k, _]). When I call k it should give me the list {k1, k2, k3}. @Mr. Wizards answer gives the list but as calling function ksym. Sort of what I need but not exactly. @halirutan answer is also interesting but I don' t want to define additional functions or operators (may be as minimum as possible).

EDIT 2

Its sort of bag or set; no order is required, only k1,k3 can be given and k2 can be missing.

EDIT 3

<< Notation`
(*below line symbolizes all subscripted k_]*)
Symbolize[ ParsedBoxWrapper[
   SubscriptBox["k", "_"]]]

somewhere here I need additional code that all k_ are added to k once its been defined. For example:

{Subscript[k, 1], Subscript[k, 2], Subscript[k, 3]} = {1, 2, 3}

after that when I ask k it should give me {Subscript[k, 1], Subscript[k, 2], Subscript[k, 3]} as @Mr.Wizard suggested ksym[] := HoldForm @@@ UpValues[k][[All, 1]] and for values kval[] := UpValues[k][[All, 2]] should work. But, what I' m missing in @Mr.Wizard example is that one needs to define symbols as;

k /: Subscript[k, 1] = "val1";
k /: Subscript[k, 2] = "val2";
k /: Subscript[k, 3] = "val3";

what I need is once I write {Subscript[k, 1], Subscript[k, 2], Subscript[k, 3]} = {1, 2, 3} I should be able to get k vector and {Subscript[k, 1], Subscript[k, 2], Subscript[k, 3]} is appended automatically to k.

Answer 1

Method #1

If you define each Subscript using TagSet⁽¹⁾⁽²⁾ the value will be associated with the Symbol, e.g. k:

k /: Subscript[k, 1] = "val1";
k /: Subscript[k, 2] = "val2";
k /: Subscript[k, 3] = "val3";

You could then use definitions such as:

kval[] := UpValues[k][[All, 2]]

ksym[] := HoldForm @@@ UpValues[k][[All, 1]]

These will always be up to date as they are dynamically generated:

kval[]

{"val1", "val2", "val3"}

ksym[]

{k₁, k₂, k₃}

Each "symbol" name is wrapped in HoldForm to prevent evaluation; ReleaseHold may be used to trigger it.

These definitions assume that no other UpValues for k are generated but if there are that can be addressed with pattern matching at the expense of speed.

---

Method #2

Following your update EDIT 3 here is another approach you might consider. Start by setting HoldFirst on Subscript so that k may be used without forming an infinite recursion, then make a definition for k that extracts the matching rules from the DownValues list:

SetAttributes[Subscript, HoldFirst]

k := Cases[DownValues@Subscript, (_[x : Subscript[k, _]] :> _) :> Hold[x]]

Make some assignments and check k:

{Subscript[k, 1], Subscript[k, 2], Subscript[k, 3]} = {"a", "b", "c"}

k

{Hold[Subscript[k, 1]], Hold[Subscript[k, 2]], Hold[Subscript[k, 3]]}

The subscripts are returned in held form, for further processing. One can use ReleaseHold to recover the values:

k // ReleaseHold

{"a", "b", "c"}

Answer 2

Although I don't like the Notation stuff very much, I regard the question as very nice, because it offers some insight into the evaluation process. When you bind UpValues to the symbol k and k has DownValues (because you want it to hold your list too), you have to make sure that k is not evaluated because otherwise the rule you gave as UpValues never kicks in.

One way to prevent this is to use an operator which holds its arguments anyway. One such an operator is += which could be interpreted in your context as append another k_n to the vector k. To keep it as simple as possible, let's say I want to following notation:

k+=1

which means, append $k_1$ to the vector $k$. If $k$ has no value, create a new list and put $k_1$ as first element.

<< Notation`
k /: k += n_ := With[{sn = ToString[n]},
  If[! ValueQ[k], k = {}];
  Symbolize[ParsedBoxWrapper[SubscriptBox["k", sn]]];
  AppendTo[k, 
   ToExpression["k\[UnderBracket]Subscript\[UnderBracket]" <> sn]]
]

And now everything happens automatically.

Table[k += i, {i, 10}];
k